Correlation between residual entropy and spanning tree entropy of ice-type models on graphs

Mikhail Isaev
School of Mathematics and Statistics
UNSW Sydney
Sydney NSW 2052, Australia
m.isaev@unsw.edu.au Supported by Australian Research Council grant DP220103074 Brendan D. McKay
School of Computing
Australian National University
Canberra ACT 2601, Australia
brendan.mckay@anu.edu.au Supported by Australian Research Council grant DP190100977 Rui-Ray Zhang
Simons Laufer Mathematical Sciences Institute
Berkeley CA 94720, USA
ruizhang@msri.org

Abstract

The logarithm of the number of Eulerian orientations, normalised by the number of vertices, is known as the residual entropy in studies of ice-type models on graphs. The spanning tree entropy depends similarly on the number of spanning trees. We demonstrate and investigate a remarkably strong, though non-deterministic, correlation between these two entropies. This leads us to propose a new heuristic estimate for the residual entropy of regular graphs that performs much better than previous heuristics. We also study the expansion properties and residual entropy of random graphs with given degrees.

1 Introduction

The graphs in this paper are free of loops but may have multiple edges. When it is not clear from the context, we will use “simple graph” or “multigraph” to emphasise that multiple edges are forbidden or allowed. An Eulerian orientation of a graph is an orientation of its edges such that every vertex has equal in-degree and out-degree.

Let $G$ be a graph with $n$ vertices and positive even degrees, and let $\operatorname{EO}(G)$ denote the number of Eulerian orientations of $G$ . We consider the logarithm of the number of Eulerian orientations of $G$ normalised by the number of vertices:

\rho(G):=\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{n}$}\log\operatorname{EO}(G).

(1.1)

If $G$ is an infinite repeating lattice of bounded degree and $\{G_{i}\}$ is an increasing sequence of Eulerian graphs which are locally like $G$ at most vertices, then under weak conditions $\rho(G_{i})$ converges to a limit $\rho(G)$ that only depends on $G$ . See [2] for a precise definition and proof. The quantity $\rho(G)$ is known as the residual entropy of ice-type models in statistical physics. Determining the asymptotic behaviour of $\rho(G)$ as $n\to\infty$ is a key question in the area, see for example [1, Chapter 8] and [18]. In particular, the value is known for the square lattice [17], the triangular lattice [1], and the hexagonal ice monolayer [16]. In addition, approximate values for many other lattice structures have been proposed, some of which we will mention below.

We can safely ignore graphs with isolated vertices. Given a degree sequence $\boldsymbol{d}=(d_{1},\ldots,d_{n})$ , define $d_{\mathrm{min}}$ , $\bar{d}$ and $d_{\mathrm{max}}$ to be the minimum, average and maximum degrees. We will also use the geometric mean $\hat{d}=(d_{1}\cdots d_{n})^{1/n}$ and the harmonic mean $d_{\mathrm{hm}}=n(d_{1}^{-1}+\cdots+d_{n}^{-1})^{-1}$ . Note that $d_{\mathrm{min}}\leqslant d_{\mathrm{hm}}\leqslant\hat{d}\leqslant\bar{d}\leqslant d_{\mathrm{max}}$ .

Around 90 years ago, Pauling [27] proposed the best-known heuristic estimate for $\rho(G)$ . Orient each edge at random. The probability that any one vertex has in-degree equal to out-degree is $2^{-d}\binom{d}{d/2}$ , where $d$ is the degree of the vertex. Assuming heuristically that these events are independent gives an estimate of $\rho(G)$ that we will call the Pauling estimate:

\widehat{\rho}(G):=\raise 0.21529pt\hbox{\small$\displaystyle\frac{1}{n}$}\,\sum_{i=1}^{n}\,\biggl{(}\log\binom{d_{i}}{d_{i}/2}-\raise 0.21529pt\hbox{\small$\displaystyle\frac{d_{i}}{2}$}\log 2\biggr{)}.

(1.2)

Lieb and Wu [18], and later Schrijver [29], showed that, for any multigraph $G$ with even degrees,

\widehat{\rho}(G)\leqslant\rho(G)\leqslant\raise 0.21529pt\hbox{\small$\displaystyle\frac{1}{2n}$}\sum_{i=1}^{n}\,\log\binom{d_{i}}{d_{i}/2}.

(1.3)

Comparing these upper and lower bounds, we find that

\rho(G)=\widehat{\rho}(G)+O(\log\hat{d}).

In this paper we make a number of theoretical and empirical contributions to the study of residual entropy. In Section 2 we first survey other known and conjectured bounds on $\rho(G)$ . For simple graphs we conjecture that in fact $\rho(G)=\widehat{\rho}(G)+o(1)$ if the harmonic mean degree goes to infinity, which will be the case if the minimum degree goes to infinity. By Theorem 2.4, this is true for sufficiently dense graphs with good expansion. Theorem 2.6 shows that $\rho(G)=\widehat{\rho}(G)+O(1)$ uniformly for all simple graphs. In Section 2.3, we report that $\rho(G)=\widehat{\rho}(G)+o(1)$ for two very broad ranges of random simple graphs with given degrees. In Section 2.4, we describe an empirical observation, supported by Theorem 2.4 and experiment, that $\rho(G)$ is highly correlated with the number of spanning trees. In combination with our knowledge of random graphs, this leads us to propose a new heuristic $\rho_{\tau}(G)$ which is a much better estimate of $\rho(G)$ than is $\widehat{\rho}(G)$ for all the graphs we have tested.

Proofs of the theorems in Section 2 are given in Sections 3 and 4. In Section 5, we explain how we computed good estimates of $\rho(G)$ even for graphs with thousands of vertices. In Section 6 we demonstrate how our new heuristic provides good estimates for products of cycles, and propose a value for the residual entropy of the simple cubic lattice.

In Section 7 we use the product of a cycle and a clique as a test case to explore the residual entropy when the degree grows as a function of the number of vertices. We find that $\rho(G)=\widehat{\rho}(G)+o(1)$ in that case, but that $\rho_{\tau}(G)$ is an even better match for $\rho(G)$ with a significantly smaller error term.

In Section 8, we show that our heuristic compares very favourably with the correct values for the triangular lattice, two types of 3-dimensional ice, and high-dimensional hypercubes.

2 Statements of the main results and conjectures

For connected regular multigraphs of degree $d\geqslant 4$ , Las Vergnas [15, Theorem 4] obtained a slightly better lower bound than (1.3) and, on condition of connectivity, a significantly better upper bound.

\operatorname{EO}(G)\leqslant K^{2/(d-2)}\bigl{(}2^{d/2g}K^{1-d/g(d-2)}\bigr{)}^{\!n},

(2.1)

where $K=2^{-d/2}\binom{d}{d/2}$ and $g$ is the girth. This bound implies that uniformly

\rho(G)\leqslant\widehat{\rho}(G)+\raise 0.21529pt\hbox{\small$\displaystyle\frac{\log d}{2g}$}+O(n^{-1}).

(2.2)

Prior to Las Vergnas’ work, a much stronger upper bound for the case of simple graphs was conjectured by Schrijver.

Conjecture 2.1 ([29]).

If a simple graph has even degrees $d_{1},\ldots,d_{n}$ , then

\operatorname{EO}(G)\leqslant\prod_{i=1}^{n}\,\operatorname{RT}(d_{i}+1)^{1/(d_{i}+1)},

where $\operatorname{RT}(d+1)$ is the number of Eulerian orientations of the complete graph (i.e., regular tournaments) with $d+1$ vertices.

If Conjecture 2.1 is correct, it is best possible for many degree sequences since it is exact for the disjoint union of complete graphs. We have computationally confirmed that no other graphs up to 12 vertices achieve the bound, and similarly for 13-vertex graphs with degrees 4 and 6, 4-regular graphs up to 19 vertices and 6-regular graphs up to 14 vertices. From [9, Theorem 5.1] we know that

\operatorname{RT}(d+1)=d^{1/2}\biggl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{2^{d+2}}{\pi(d+1)}$}\biggr{)}^{\!d/2}e^{-\frac{1}{2}+O(d^{-1})},

(2.3)

which enables us to calculate (since we disallow isolated vertices) that Conjecture 2.1 would imply that

\rho(G)\leqslant\widehat{\rho}(G)+\raise 0.21529pt\hbox{\small$\displaystyle\frac{1}{n}$}\sum_{i=1}^{n}\,\raise 0.21529pt\hbox{\small$\displaystyle\frac{O(1)+\log d_{i}}{d_{i}}$}\leqslant\widehat{\rho}(G)+\raise 0.21529pt\hbox{\small$\displaystyle\frac{O(1)+\log d_{\mathrm{hm}}}{d_{\mathrm{hm}}}$}

for simple graphs, where the second inequality follows by the concavity of the function $x\log(1/x)$ for $x\geqslant 0$ .

Though we don’t know how to prove Conjecture 2.1, our evidence suggests that at least the following implication is true.

Conjecture 2.2.

If $G=G(n)$ is a sequence of simple graphs with even degrees $\boldsymbol{d}=\boldsymbol{d}(n)$ , such that the harmonic mean degree $d_{\mathrm{hm}}\to\infty$ as $n\to\infty$ , then

\rho(G)=\widehat{\rho}(G)+o(1).

Recall that $d_{\mathrm{min}}\leqslant d_{\mathrm{hm}}\leqslant\hat{d}\leqslant\bar{d}\leqslant d_{\mathrm{max}}$ . The condition $d_{\mathrm{hm}}\to\infty$ is implied by $d_{\mathrm{min}}\to\infty$ , but the weaker condition $\hat{d}\to\infty$ is not sufficient. For $n$ being an odd multiple of 6, define $G_{n}$ to be the disjoint union of $K_{n/2}$ and $n/6$ triangles. Then the geometric mean degree is $\hat{d}\sim\sqrt{n}$ , but $\operatorname{EO}(G_{n})=2^{n/6}\operatorname{RT}(n/2)$ , which implies by (2.3) that $\rho(G_{n})-\widehat{\rho}(G_{n})\to\frac{1}{6}\log 2$ . Note that the harmonic mean degree is $4+o(1)$ .

The converse of Conjecture 2.2 is also not true. Even $d_{\mathrm{max}}=O(1)$ is insufficient to imply $\rho(G)=\widehat{\rho}(G)+\Omega(1)$ , as shown by the case of increasing girth in (2.2). Another observation is that if $G_{1},G_{2},\ldots$ is any sequence of graphs such that $\rho(G_{i})=\widehat{\rho}(G_{i})+o(1)$ , then the same is true for $G^{\prime}_{1},G^{\prime}_{2},\ldots$ , where $G^{\prime}_{i}$ is $G_{i}$ with any number of edges subdivided, even though the average degree may approach 2.

It suffices to prove Conjecture 2.2 for connected simple graphs, on account of the following lemma whose proof will appear in Section 3.

Lemma 2.3.

Let $\mathcal{C}$ be a class of connected simple graphs for which Conjecture 2.2 holds. Then the conjecture also holds for graphs whose components all lie in $\mathcal{C}$ .

2.1 Sufficiently growing degrees and good expansion

A recent result of the present authors shows that Conjecture 2.2 holds for simple graphs with good expansion properties and sufficiently high degrees. Recall that the isoperimetric number (also known as the Cheeger constant) of a graph $G$ is defined by

h(G):=\min\biggl{\{}\raise 0.21529pt\hbox{\small$\displaystyle\frac{|\partial_{G}\,U|}{|U|}$}\mathrel{:}U\subset V(G),1\leqslant|U|\leqslant\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{2}$}|V(G)|\biggr{\}},

(2.4)

where $\partial_{G}\,U$ is the set of edges of $G$ with one end in $U$ and the other end in $V(G)\setminus U$ . Note that $h(G)\leqslant d_{\mathrm{min}}(G)$ .

Theorem 2.4 (Isaev, McKay, Zhang [9]).

Let $G=G(n)$ be a simple graph with $n$ vertices and even degrees. Assume that $d_{\mathrm{max}}\gg\log^{8}n$ and $h(G)\geqslant\gamma d_{\mathrm{max}}$ for some constant $\gamma>0$ . Then,

\operatorname{EO}(G)=\raise 0.21529pt\hbox{\small$\displaystyle\frac{2^{|E(G)|}}{\sqrt{t(G)}}$}\,\biggl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{2}{\pi}$}\biggr{)}^{(n-1)/2}\!\!\exp\biggl{(}-\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{4}$}\sum_{jk\in G}\biggl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{1}{d_{j}}$}+\raise 0.21529pt\hbox{\small$\displaystyle\frac{1}{d_{k}}$}\biggr{)}^{\!2\,}+O\biggl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{n}{d_{\mathrm{max}}^{2}}$}\log\raise 0.21529pt\hbox{\small$\displaystyle\frac{2n}{d_{\mathrm{max}}}$}\biggr{)}\biggr{)},

where $t(G)$ is the number of spanning trees of $G$ .

In relation to Conjecture 2.2, the following consequence of Theorem 2.4 is proved in Section 3.

Corollary 2.5.

Under the assumptions of Theorem 2.4, we have that

\rho(G)=\widehat{\rho}(G)+O\mathopen{}\mathclose{{}\left(\raise 0.21529pt\hbox{\small$\displaystyle\frac{\log^{2}d_{\mathrm{min}}}{d_{\mathrm{min}}}$}}\right).

2.2 A new upper bound on $\operatorname{EO}(G)$

Our next contribution is a new upper bound on $\rho(G)$ , which implies that $\rho(G)=\widehat{\rho}(G)+O(1)$ for all simple graphs.

Theorem 2.6.

For any connected multigraph $G$ with even degrees and $t(G)$ spanning trees,

\operatorname{EO}(G)\leqslant\raise 0.21529pt\hbox{\small$\displaystyle\frac{2^{\lvert E(G)\rvert+3(n-1)/2}}{\pi^{(n-1)/2}\,t(G)^{1/2}}$}.

Corollary 2.7.

For simple graphs $G$ with even degrees,

\widehat{\rho}(G)\leqslant\rho(G)\leqslant\widehat{\rho}(G)+\begin{cases}\,\lower 0.51663pt\hbox{\large$\textstyle\frac{27}{10}$},&\text{~{}always;}\\[2.15277pt] \,\lower 0.51663pt\hbox{\large$\textstyle\frac{21}{22}$},&\text{~{}if $G$ is regular;}\\[2.15277pt] \,\lower 0.51663pt\hbox{\large$\textstyle\frac{5}{6}$},&\text{~{}if $G$ is regular without 3-cycles.}\end{cases}

Moreover, if $d_{\mathrm{min}}=d_{\mathrm{min}}(n)\to\infty$ ,

\widehat{\rho}(G)\leqslant\rho(G)\leqslant\widehat{\rho}(G)+\log 2+O\Bigl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{\log^{2}d_{\mathrm{min}}}{d_{\mathrm{min}}}$}\Bigr{)}.

The above theorem and its corollary are proved in Section 3. If Conjecture 2.1 is correct, the largest value of $\rho(G)-\widehat{\rho}(G)$ for simple graphs is actually $\frac{1}{3}\log 2\approx 0.23105$ , occurring for disjoint unions of triangles. Note that, although Theorem 2.6 holds for multigraphs, Corollary 2.7 does not. For multigraphs, the upper bound in (1.3) is achieved, so $\rho(G)-\widehat{\rho}(G)$ is not uniformly bounded.

2.3 Random graphs with given degrees

Let $\mathcal{G}(n,\boldsymbol{d})$ denote the uniform probability space of simple graphs with $n$ vertices and degree sequence $\boldsymbol{d}$ . We will prove that Conjecture 2.2 holds in the following probabilistic sense. The proof will appear in Section 4.

Theorem 2.8.

Let $\boldsymbol{d}=(d_{1},\ldots,d_{n})$ be a graphical degree sequence such that each $d_{i}$ is positive and even and either of the following two conditions holds:

(R1)

$d_{\mathrm{max}}^{2}=o(n)$ ,
(R2)

$d_{\mathrm{max}}\gg\log^{8}n$ and $d_{\mathrm{min}}\geqslant\gamma d_{\mathrm{max}}$ for some fixed $\gamma>0$ .

If $G\sim\mathcal{G}(n,\boldsymbol{d})$ then, for any fixed $\varepsilon>0$ ,

\operatorname{\mathbb{P}}\bigl{(}\rho(G)>\widehat{\rho}(G)+\varepsilon\bigr{)}\leqslant e^{-\Omega(d_{\mathrm{max}}^{2}+n)}.

Theorem 2.8 follows immediately from two more detailed results Theorem 4.2 and Theorem 4.4 that explore the dependence of the probability bounds with respect to $\varepsilon$ . In particular, these results show that if $G\sim\mathcal{G}(n,\boldsymbol{d})$ then, with probability tending to $1$ ,

•

$\rho(G)\leqslant\widehat{\rho}(G)+O\mathopen{}\mathclose{{}\left(\lower 0.51663pt\hbox{\large$\textstyle\frac{d_{\mathrm{max}}^{2}+\log n}{n}$}}\right)$ for the range (R1) of Theorem 2.8;
•

$\rho(G)\leqslant\widehat{\rho}(G)+O\mathopen{}\mathclose{{}\left(\lower 0.51663pt\hbox{\large$\textstyle\frac{\log^{2}d_{\mathrm{max}}}{d_{\mathrm{max}}}$}}\right)$ for the range (R2) of Theorem 2.8.

For the range (R1), we combine the result of [21] on the enumeration of bipartite graphs with the switching method to find an asymptotic formula for

\operatorname{\mathbb{E}}\,[\operatorname{EO}(G)]=\operatorname{\mathbb{E}}\,[e^{n\rho(G)}].

It turns out that

\operatorname{\mathbb{E}}\,[e^{n\rho(G)}]=e^{n\widehat{\rho}(G)+O(d_{\mathrm{max}}^{2})}.

Then we use standard arguments to show the concentration of $\rho(G)$ .

For the range (R2), we employ Theorem 2.4. Note that it requires $G$ to have a sufficiently large isoperimetric constant. Applying the switching method, we show that random graphs with given degrees are good expanders with high probability, which could be of independent interest; see Section 4.1.

2.4 A new heuristic estimate for regular graphs

The asymptotic formula in Theorem 2.4 suggests that $\rho(G)+\frac{1}{2}\tau(G)$ may exhibit much less dependency than $\rho(G)$ on the structure of the graph. Here the spanning tree entropy $\tau(G)$ is the logarithm of the number of spanning trees of $G$ normalised by the number of vertices:

\tau(G):=\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{n}$}\log t(G).

As a consequence of Theorem 2.8 we have established that Pauling’s estimate $\widehat{\rho}(G)=\log\binom{d}{d/2}-\lower 0.51663pt\hbox{\large$\textstyle\frac{d}{2}$}\log 2$ is asymptotically correct for random simple $d$ -regular graphs provided $d(n)$ grows quickly enough. McKay [20] showed that spanning tree entropy for this random graph model (for the case when $3\leqslant d=O(1)$ ) is concentrated around

\tau_{d}:=\log\raise 0.21529pt\hbox{\small$\displaystyle\frac{(d-1)^{d-1}}{(d^{2}-2d)^{d/2-1}}$}.

Thus, it is natural to consider the following estimate for the residual entropy of a simple $d$ -regular graph based on a correction of the number of spanning trees with respect to a random graph:

\rho_{\tau}(G):=\widehat{\rho}(G)+\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{2}$}\tau_{d}-\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{2}$}\tau(G).

(2.5)

It follows from [20, Thm. 5.2] that $\tau(G)<\tau_{d}(G)$ if $G$ is $d$ -regular for $d\geqslant 4$ , except possibly for $d=4,n\leqslant 18$ . Thus, our estimate is consistent with $\rho(G)\geqslant\widehat{\rho}(G)$ .

Refer to caption — Figure 1: $\rho(G)$ (vertical) versus $\tau(G)$ (horizontal) for some randomised graphs. The solid line is $\rho_{\tau}(G)$ .

In order to illustrate our case for this estimate, we tested several hundred large graphs of degree 4 or 6. The method by which $\rho(G)$ was estimated will be described in Section 5. To show a continuum between a regular lattice structure and a random graph, we started with a 2-dimensional square lattice $C_{40}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{40}$ (1600 vertices, degree 4), and a 3-dimensional simple cubic lattice $C_{12}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{12}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{12}$ (1728 vertices, degree 6) and applied the switching $\{ab,cd\}\to\{ac,cd\}$ in random places between 0 and 10,000 times (avoiding multiple edges and loops). The results shown in Figure 1 suggest that the correlation between $\rho(G)$ and $\tau(G)$ is even stronger than between $\rho(G)$ and $\rho_{\tau}(G)$ . However, the relationship between the two entropies is not exact, as shown in Figure 2.

The next theorem shows that even a weaker asymptotic relationship fails to hold in general. This is unfortunate as such a relationship would be sufficient to demonstrate that the residual entropies of hexagonal ice (Ih) and cubic ice (Ic) are identical; see Section 8.2.

Theorem 2.9.

There is no function $\mathring{\rho}(d,\tau)$ with the following property for every $d$ and $\tau$ . If $G_{1},G_{2},\ldots$ is an increasing sequence of connected $d$ -regular graphs such that $\tau(G_{i})\to\tau$ , then $\rho(G_{i})\to\mathring{\rho}(d,\tau)$ .

Proof.

(See Section 6 for the definitions and elementary theory.) For $m\geqslant 3$ , define the Cartesian products $H_{1}(m):=G_{1}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{m}$ and $H_{2}(m):=G_{2}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{m}$ , where $G_{1}$ and $G_{2}$ are the graphs in Figure 2 and $C_{m}$ is the cycle of length $m$ . Since $H_{1}(m)$ and $H_{2}(m)$ have the same eigenvalues, they have the same spanning tree entropy and it converges to a limit as $m\to\infty$ . However, using the transfer matrix method (see Theorem 6.3 below), we have determined that

\lim_{m\to\infty}\rho(H_{1}(m))\approx 0.9525279,\qquad\lim_{m\to\infty}\rho(H_{2}(m))\approx 0.9524817.

Since the two limits are different, $\mathring{\rho}(6,\tau)$ doesn’t exist. ∎

3 Proof of Theorem 2.6, Corollary 2.5 and Lemma 2.3

The Laplacian matrix $L(G)$ of a loop-free multigraph $G$ with $n$ vertices is the $n\times n$ matrix with entries given by

L_{ij}=\begin{cases}\,d_{i},&\text{ if $i=j$ and $d_{i}$ is the degree of vertex $i$};\\ \,-\ell,&\text{ if $i\neq j$ and $\ell$ edges join vertices $i$ and $j$}.\end{cases}

As is well known, the number of zero eigenvalues of $L(G)$ equals the number of components of $G$ , and the other eigenvalues are strictly positive. In addition, the Matrix Tree Theorem says that the number of spanning trees of $G$ is the absolute value of every minor of $L(G)$ .

Proof of Theorem 2.6..

The number of Eulerian orientations of $G$ is the constant term in

\prod_{jk\in G}\,\Bigl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{x_{j}}{x_{k}}$}+\raise 0.21529pt\hbox{\small$\displaystyle\frac{x_{k}}{x_{j}}$}\Bigr{)}.

As shown in [8], this implies that

\operatorname{EO}(G)=2^{\lvert E(G)\rvert}\pi^{-n}\int_{-\pi/2}^{\pi/2}\!\!\!\cdots\!\int_{-\pi/2}^{\pi/2}F(\boldsymbol{\theta})\,d\boldsymbol{\theta},

where $\boldsymbol{\theta}=(\theta_{1},\ldots,\theta_{n})$ and

F(\boldsymbol{\theta})=\prod_{jk\in G}\cos(\theta_{j}-\theta_{k}).

Since $F(\boldsymbol{\theta})$ is invariant under uniform shift of the arguments, we can fix $\theta_{n}=0$ and write

\operatorname{EO}(G)=2^{\lvert E(G)\rvert}\pi^{-n+1}\int_{-\pi/2}^{\pi/2}\!\!\!\cdots\!\int_{-\pi/2}^{\pi/2}F(\theta_{1},\ldots,\theta_{n-1},0)\,d\theta_{1}\cdots d\theta_{n-1}.

Define $\boldsymbol{\phi}=(\phi_{1},\ldots,\phi_{n-1})$ , where

\phi_{j}=\begin{cases}\,\frac{\pi}{2}-\theta_{j},&~{}\text{ if $\frac{\pi}{4}\leqslant\theta_{j}\leqslant\frac{\pi}{2}$};\\ \;\theta_{j},&~{}\text{ if $-\frac{\pi}{4}\leqslant\theta_{j}\leqslant\frac{\pi}{4}$};\\ \,-\frac{\pi}{2}-\theta_{j},&~{}\text{ if $-\frac{\pi}{2}\leqslant\theta_{j}\leqslant-\frac{\pi}{4}$},\end{cases}

and also set $\phi_{n}=0$ for convenience. Note that $\boldsymbol{\theta}\in\bigl{[}-\frac{\pi}{2},\frac{\pi}{2}\bigr{]}^{n}$ implies $\boldsymbol{\phi}\in\bigl{[}-\frac{\pi}{4},\frac{\pi}{4}\bigr{]}^{n}$ . By considering all the cases, we can check that, for $\theta_{j},\theta_{k}\in\bigl{[}-\frac{\pi}{2},\frac{\pi}{2}\bigr{]}$ ,

\lvert\phi_{j}-\phi_{k}\rvert\leqslant\min\bigl{\{}\lvert\theta_{j}-\theta_{k}\rvert,\pi-\lvert\theta_{j}-\theta_{k}\rvert\bigr{\}}.

We also have, for $x\in[-\pi,\pi]$ , that $\lvert\cos x\rvert\leqslant\exp\bigl{(}-\frac{1}{2}\min\{\lvert x\rvert,\pi-\lvert x\rvert\}^{2}\bigr{)}$ . Therefore, for $\boldsymbol{\theta}\in\bigl{[}-\frac{\pi}{2},\frac{\pi}{2}\bigr{]}^{n}$ , we have

\lvert F(\boldsymbol{\theta})\rvert\leqslant\exp\Bigl{(}-\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{2}$}\sum_{jk\in G}(\phi_{j}-\phi_{k})^{2}\Bigr{)}=\exp\Bigl{(}-\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{2}$}\boldsymbol{\phi}^{\mathrm{T}\!}Q\boldsymbol{\phi}\Bigr{)},

where $Q=Q(G)$ is the Laplacian matrix of $G$ with the final row and column removed. In changing variables from $\boldsymbol{\theta}$ to $\boldsymbol{\phi}$ in the integral, we need to take account of the fact that $\theta_{j}\mapsto\phi_{j}$ is a two-to-one map. To compensate for this, we multiply by $2^{n-1}$ :

	$\displaystyle\operatorname{EO}(G)$	$\displaystyle\leqslant 2^{\lvert E(G)\rvert+n-1}\pi^{-n+1}\int_{-\pi/4}^{\pi/4}\!\!\cdots\!\int_{-\pi/4}^{\pi/4}\exp\Bigl{(}-\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{2}$}\boldsymbol{\phi}^{\mathrm{T}\!}Q\boldsymbol{\phi}\Bigr{)}\,d\boldsymbol{\phi}$
		$\displaystyle\leqslant 2^{\lvert E(G)\rvert+n-1}\pi^{-n+1}\int_{-\infty}^{\infty}\!\!\cdots\!\int_{-\infty}^{\infty}\exp\Bigl{(}-\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{2}$}\boldsymbol{\phi}^{\mathrm{T}\!}Q\boldsymbol{\phi}\Bigr{)}\,d\boldsymbol{\phi}$
		$\displaystyle=2^{\lvert E(G)\rvert+3(n-1)/2}\pi^{-(n-1)/2}\lvert Q\rvert^{-1/2}$
		$\displaystyle=2^{\lvert E(G)\rvert+3(n-1)/2}\pi^{-(n-1)/2}t(G)^{-1/2},$

where the final step is to apply the Matrix Tree Theorem. ∎

Lemma 3.1 (Kostochka [12]).

A connected simple graph $G$ with $n$ vertices and minimum degree $d_{\mathrm{min}}\geqslant 2$ has

\bigl{(}\hat{d}\,d_{\mathrm{min}}^{-6\log d_{\mathrm{min}}/d_{\mathrm{min}}}M(d_{\mathrm{min}})^{-1}\bigr{)}^{n}\leqslant t(G)\leqslant\raise 0.21529pt\hbox{\small$\displaystyle\frac{1}{n-1}$}\hat{d}^{\,n},

where $\hat{d}$ is the geometric mean degree and $M(d_{\mathrm{min}})=\min\{2,d_{\mathrm{min}}^{3\log d_{\mathrm{min}}/d_{\mathrm{min}}}\}$ .

Proof.

This is not stated explicitly in [12] but follows from the proof given there plus the observation that

\sum_{i=0}^{\lfloor 3n\log k/k\rfloor}\binom{n-1}{i}\leqslant\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{2}$}\min\{2,k^{3\log k/k}\}^{n}

for $2\leqslant k\leqslant n-1$ . Note that the constants in Kostochka’s proof are far from optimized. Improving them would also improve the constants in our Corollary 2.7, but we have not attempted to do that. ∎

Proof of Corollary 2.7.

Suppose $G$ is disconnected, with components $G_{1},\ldots,G_{m}$ . Let $n_{i}$ be the number of vertices of $G_{i}$ , and let $n=n_{1}+\cdots+n_{m}$ . From the definition of $\widehat{\rho}(G)$ , and the fact that $\operatorname{EO}(G)=\prod_{i=1}^{m}\operatorname{EO}(G_{i})$ , we have

\rho(G)-\widehat{\rho}(G)=\sum_{i=1}^{m}\,\raise 0.21529pt\hbox{\small$\displaystyle\frac{n_{i}}{n}$}\bigl{(}\rho(G_{i})-\widehat{\rho}(G_{i})\bigr{)}\leqslant\max\nolimits_{i=1}^{m}\bigl{(}\rho(G_{i})-\widehat{\rho}(G_{i})\bigr{)}.

(3.1)

Therefore it suffices to prove Corollary 2.7 for connected graphs $G$ .

For $d\geqslant 2$ , asymptotic expansion for large $d$ and computation for small $d$ gives

\log\binom{d}{d/2}\geqslant d\log 2-\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{2}$}\log d-\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{2}$}\log\raise 0.21529pt\hbox{\small$\displaystyle\frac{\pi}{2}$}-\raise 0.21529pt\hbox{\small$\displaystyle\frac{1}{4d}$}.

(3.2)

Define $\rho_{\mathrm{new}}(G)$ to be the upper bound on $\rho(G)$ given by Theorem 2.6 in conjunction with Lemma 3.1. Then

$\displaystyle\rho_{\mathrm{new}}(G)-\rho(G)$	$\displaystyle\leqslant\raise 0.21529pt\hbox{\small$\displaystyle\frac{1}{n}$}\sum_{i=0}^{n}\delta(d_{i},d_{\mathrm{min}},n)\leqslant\max\nolimits_{i=1}^{n}\delta(d_{i},d_{\mathrm{min}},n),\quad\text{where}$
$\displaystyle\delta(d_{i},d_{\mathrm{min}},n)$	$\displaystyle=d_{i}\log 2+2\log 2-\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{2}$}\log\pi-\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{2}$}\log d_{i}-\log\binom{d_{i}}{d_{i}/2}$
	$\displaystyle{\qquad}+\raise 0.21529pt\hbox{\small$\displaystyle\frac{3}{d_{\mathrm{min}}}$}\log^{2}d_{\mathrm{min}}+\raise 0.21529pt\hbox{\small$\displaystyle\frac{1}{2n}$}\log\raise 0.21529pt\hbox{\small$\displaystyle\frac{\pi}{8}$}$	(3.3)
	$\displaystyle\leqslant\lower 0.51663pt\hbox{\large$\textstyle\frac{3}{2}$}\log 2+\raise 0.21529pt\hbox{\small$\displaystyle\frac{1}{4d_{\mathrm{min}}}$}+\raise 0.21529pt\hbox{\small$\displaystyle\frac{3}{d_{\mathrm{min}}}$}\log^{2}d_{\mathrm{min}},$	(3.4)

where we have applied $M(d_{\mathrm{min}})\leqslant 2$ , (3.2), $\log\frac{\pi}{8}<0$ and $d_{i}\geqslant d_{\mathrm{min}}$ . The expression (3.4) is unimodal, with its largest value for even $d_{\mathrm{min}}$ at $d_{\mathrm{min}}=8$ and other values less than 2.69. The value of (3.3) for $d_{i}=d_{\mathrm{min}}=8$ and $n\to\infty$ is less than $2.69242<\frac{27}{10}$ .

In the case of regular graphs, the second claim is trivial if the degree $d=2$ so assume $d\geqslant 4$ . For $4\leqslant d\leqslant 760$ in the second case, and $4\leqslant d\leqslant 1900$ in the third case, the bounds follow from (2.1), noting that the worst case for $n$ is $n=d+1$ . For larger degrees, apply Theorem 2.6 and Lemma 3.1 with $M(d_{\mathrm{min}})\leqslant d_{\mathrm{min}}^{3\log d_{\mathrm{min}}/d_{\mathrm{min}}}$ .

The last claim follows from (3.4). ∎

Proof of Lemma 2.3.

Suppose $G$ has components $G_{1},\ldots,G_{m}$ with orders $\alpha_{1}n,\ldots,\alpha_{m}n$ . Define

g(h)=\sup\bigl{\{}\rho(H)-\widehat{\rho}(H)\mathrel{:}d_{\mathrm{hm}}(H)\geqslant h,H\in\mathcal{C}\bigr{\}}.

Clearly $g(h)$ is non-increasing, and by the assumptions of the lemma, $g(h)\to 0$ as $h\to\infty$ . Suppose $d_{\mathrm{hm}}(G)\to\infty$ . Without loss of generality, for some $\ell\geqslant 0$ , $d_{\mathrm{hm}}(G_{i})\leqslant d_{\mathrm{hm}}(G)^{1/2}$ for $1\leqslant i\leqslant\ell$ and $d_{\mathrm{hm}}(G_{i})>d_{\mathrm{hm}}(G)^{1/2}$ for $\ell+1\leqslant i\leqslant m$ . By the definition of harmonic mean, $d_{\mathrm{hm}}(G)^{-1}=\sum_{i=1}^{m}\alpha_{i}d_{\mathrm{hm}}(G_{i})^{-1}\geqslant\sum_{i=1}^{\ell}\alpha_{i}d_{\mathrm{hm}}(G)^{-1/2}$ , so $\sum_{i=1}^{\ell}\alpha_{i}\leqslant d_{\mathrm{hm}}(G)^{-1/2}$ . Now, by (3.1) and Corollary 2.7,

\rho(G)-\widehat{\rho}(G)=\sum_{i=1}^{m}\alpha_{i}(\rho(G_{i})-\widehat{\rho}(G_{i}))\leqslant\lower 0.51663pt\hbox{\large$\textstyle\frac{27}{10}$}\sum_{i=1}^{\ell}\alpha_{i}+\sum_{i=\ell+1}^{m}\alpha_{i}g(d_{\mathrm{hm}}(G)^{1/2})=o(1).\qed

Proof of Corollary 2.5.

Observe that definition (2.4) and assumption $h(G)\geqslant\gamma d_{\mathrm{max}}$ imply that $d_{\mathrm{min}}\geqslant\gamma d_{\mathrm{max}}$ . Applying Theorem 2.4 and using Lemma 3.1 and the asymptotic formula (3.2) we obtain the claimed bound. ∎

4 Proof of Theorem 2.8

We first prove the case of $d_{\mathrm{max}}=o(\sqrt{n})$ .

Lemma 4.1.

Let $\boldsymbol{d}=\boldsymbol{d}(n)=(d_{1},\ldots,d_{n})$ be a degree sequence with all degrees positive and even. Assume $d_{\mathrm{max}}^{2}=o(\bar{d}n)$ . Then the average number of Eulerian orientations of a random undirected simple graph with degree sequence $\boldsymbol{d}$ is

n^{1/2}\,2^{-\bar{d}n/2}e^{O(d_{\mathrm{max}}^{2})}\prod_{i=1}^{n}\binom{d_{i}}{d_{i}/2}.

Proof.

The average is equal to the number of Eulerian oriented graphs with in-degrees and out-degrees $\frac{1}{2}\boldsymbol{d}$ , divided by the number of undirected graphs with degrees $\boldsymbol{d}$ .

A digraph with in-degrees and out-degrees $\frac{1}{2}\boldsymbol{d}$ can be represented as an undirected bipartite graph with vertices $v_{1},\ldots,v_{n}$ and $w_{1},\ldots,w_{n}$ . For each $i$ , both $v_{i}$ and $w_{i}$ have degree $d_{i}/2$ . Vertex $v_{i}$ is adjacent to $w_{j}$ if the digraph has an edge from vertex $i$ to vertex $j$ . Since the digraph has no loops, $v_{i}$ is not adjacent to $w_{i}$ for any $i$ . With such restrictions, the number of bipartite graphs given in [21, Theorem 4.6] is, to the precision we need here,

\raise 0.21529pt\hbox{\small$\displaystyle\frac{(\bar{d}n/2)!}{\bigl{(}\prod_{i=1}^{n}(d_{i}/2)!\bigr{)}^{2}}$}\,e^{O(d_{\mathrm{max}}^{2})}.

(4.1)

However, such bipartite graphs may correspond to digraphs with 2-cycles, which is not permitted for simple Eulerian oriented graphs. This occurs if, for some $i\neq j$ , $v_{i}$ is adjacent to $w_{j}$ , and $v_{j}$ is adjacent to $w_{i}$ . Arbitrarily order all such potential bad pairs and label the corresponding events by $E_{1},\ldots,E_{\binom{n}{2}}$ . The probability that none of these events occurs is the product of conditional probabilities:

P_{\boldsymbol{d}}=\prod_{i=1}^{\binom{n}{2}}\,\bigl{(}1-\operatorname{\mathbb{P}}(E_{i}\mid\cap_{1\leqslant j<i}\bar{E}_{j})\bigr{)},

(4.2)

where $\bar{E}_{j}$ is the complement of $E_{j}$ . Consider the bad event $E_{x}$ that $\{v_{i},w_{j}\}$ and $\{v_{j},w_{i}\}$ are both edges. Choose two additional edges $\{v^{\prime},w^{\prime}\}$ and $\{v^{\prime\prime},w^{\prime\prime}\}$ , and replace these four edges with $\{v_{i},w^{\prime}\}$ , $\{v_{j},w^{\prime\prime}\}$ , $\{v^{\prime},w_{i}\}$ and $\{v^{\prime\prime},w_{j}\}$ . Provided $v^{\prime}\neq v^{\prime\prime}$ , $w^{\prime}\neq w^{\prime\prime}$ and none of the additional edges are incident to any of $v_{i},v_{j},w_{i},w_{j}$ or their neighbours, this operation removes the bad pair of edges $\{v_{i},w_{j}\}$ and $\{v_{j},w_{i}\}$ without creating any additional bad pairs. Observe that it can be performed in $\frac{1}{4}n^{2}\bar{d}^{2}-O(nd_{\mathrm{max}}^{2}\bar{d})=\Omega(n^{2}\bar{d}^{2})$ ways and doesn’t change the degree sequence.

In the reverse direction, if the event $E_{x}$ does not occur and we wish to create it, the operation is determined by the choice of an edge incident with each of $v_{i},v_{j},w_{i},w_{j}$ , which can be made in $O(d_{\mathrm{max}}^{4})$ ways. Note that both these bounds hold irrespective of any conditioning on other bad events not occurring. Therefore, the conditional probability of this bad event occurring is $O\bigl{(}d_{\mathrm{max}}^{4}/(n^{2}\bar{d}^{2})\bigr{)}$ . By (4.2),

P_{\boldsymbol{d}}=\bigl{(}1-O(d_{\mathrm{max}}^{4}/(n^{2}\bar{d}^{2}))\bigr{)}^{\binom{n}{2}}=e^{-O(d_{\mathrm{max}}^{2})}.

Thus we find that the number of Eulerian oriented graphs with in-degrees and out-degrees $\frac{1}{2}\boldsymbol{d}$ is also given by (4.1).

The number of undirected simple graphs with degrees $\boldsymbol{d}$ was determined in [22, Theorem 4.6] to be

\raise 0.21529pt\hbox{\small$\displaystyle\frac{(\bar{d}n)!}{(\bar{d}n/2)!\,2^{\bar{d}n/2}\prod_{i=1}^{n}d_{i}!}$}\,e^{O(d_{\mathrm{max}}^{2})}.

(4.3)

Dividing (4.1) by (4.3) completes the proof. ∎

Theorem 4.2.

Consider $G\sim\mathcal{G}(n,\boldsymbol{d})$ , where $\boldsymbol{d}$ satisfies the assumptions of Lemma 4.1. Then, for any $\mu=\mu(n)>0$ ,

\operatorname{\mathbb{P}}\bigl{(}\rho(G)\geqslant\widehat{\rho}(G)+\mu\bigr{)}\leqslant\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{{e^{n\mu}-1}}$}n^{1/2}e^{O(d_{\mathrm{max}}^{2})}.

Proof.

Let $p=\operatorname{\mathbb{P}}\bigl{(}\rho(G)\geqslant\widehat{\rho}(G)+\mu\bigr{)}$ . By (1.3), we have $\rho(G)\geqslant\widehat{\rho}(G)$ always. Then

\operatorname{\mathbb{E}}\,[\operatorname{EO}(G)]=\operatorname{\mathbb{E}}\,[e^{n\rho(G)}]\geqslant(1-p)e^{n\widehat{\rho}(G)}+pe^{n(\widehat{\rho}(G)+\mu)}=(1-p+e^{n\mu}p)\,e^{n\widehat{\rho}(G)}.

Comparing this to the estimate of the expectation in Lemma 4.1, we have

(1-p+e^{n\mu}p)\,e^{n\widehat{\rho}(G)}\leqslant n^{1/2}\,2^{-\bar{d}n/2}e^{O(d_{\mathrm{max}}^{2})}\prod_{i=1}^{n}\binom{d_{i}}{d_{i}/2},

and by the definition of $\widehat{\rho}(G)$ in (1.2),

p\leqslant\raise 0.21529pt\hbox{\small$\displaystyle\frac{1}{{e^{n\mu}-1}}$}\biggl{(}n^{1/2}\,2^{-\bar{d}n/2}e^{O(d_{\mathrm{max}}^{2})}\prod_{i=1}^{n}\binom{d_{i}}{d_{i}/2}-2^{-\bar{d}n/2}\prod_{i=1}^{n}\binom{d_{i}}{d_{i}/2}\biggr{)},

which completes the proof. ∎

4.1 Expansion properties of random graphs with given degrees

The next theorem establishes a lower bound on the isoperimetric constant of a random graph with given degrees. Previous results have been restricted to random regular graphs, see [4, 13, 11].

Theorem 4.3.

Let $\boldsymbol{d}=\boldsymbol{d}(n)$ be a degree sequence and $C=C(n)>0$ be such that we have $d_{\mathrm{min}}\geqslant 24+32C$ and $d_{\mathrm{max}}^{2}\leqslant C\bar{d}n$ , where $\bar{d}$ is the average degree. Define $\alpha:=\frac{1}{6+8C}$ . If $G\sim\mathcal{G}(n,\boldsymbol{d})$ , then

\operatorname{\mathbb{P}}\bigl{(}h(G)\geqslant\alpha d_{\mathrm{min}}\bigr{)}\geqslant 1-\Bigl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{d_{\mathrm{min}}^{2}}{5\bar{d}n}$}\Bigr{)}^{\frac{5}{26}\alpha d_{\mathrm{min}}^{2}}.

Proof.

Let $S\subseteq\{1,\ldots,n\}$ , and let $S_{k}$ be the set of all graphs with degree sequence $\boldsymbol{d}$ such that $k$ is the number of edges between $S$ and $\bar{S}:=\{1,\ldots,n\}-S$ . Assume that $s:=|S|\leqslant\frac{1}{2}n$ and $k\leqslant 2\alpha d_{\mathrm{min}}s$ . We can also assume that $s\geqslant(1-\alpha)d_{\mathrm{min}}$ , since otherwise at least $\alpha d_{\mathrm{min}}s$ edges leave $S$ .

Consider $G_{k}\in S_{k}$ . We can create a graph in $S_{k+2}$ by removing two edges $v_{1}v_{2}\in G_{k}[S]$ , an $w_{1}w_{2}\in G_{k}[\bar{S}]$ and replacing them by either $v_{1}w_{1}$ and $v_{2}w_{2}$ , or $v_{1}w_{2}$ and $v_{2}w_{1}$ . One or both of these operations may be unavailable due to an existing edge. If $d^{\prime}$ is the average degree of a vertex in $S$ , the number of ways to perform the operation is at least

$\displaystyle\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{2}$}(d^{\prime}s-k)(\bar{d}n-d^{\prime}s-k)-kd_{\mathrm{max}}^{2}$	$\displaystyle\geqslant\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{2}$}(d_{\mathrm{min}}s-k)(\bar{d}n-d_{\mathrm{min}}s-k)-Ck\bar{d}n$
	$\displaystyle\geqslant\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{4}$}(1-4\alpha+4\alpha^{2}-8\alpha C)d_{\mathrm{min}}\bar{d}sn$
	$\displaystyle=\raise 0.21529pt\hbox{\small$\displaystyle\frac{1+C}{(3+4C)^{2}}$}d_{\mathrm{min}}\bar{d}sn,$	(4.4)

where the first inequality holds since the first expression is increasing in $d^{\prime}$ for $s\leqslant\frac{1}{2}n$ and $d_{\mathrm{max}}^{2}\leqslant C\bar{d}n$ , while the second line holds since $k\leqslant 2\alpha d_{\mathrm{min}}s$ and $s\leqslant\frac{1}{2}n$ .

Given $G_{k+2}\in S_{k+2}$ we can recover a graph in $G_{k}$ by performing the same operation in reverse, which is determined by the choice of two edges between $S$ and $\bar{S}$ . This time we need an upper bound, namely

\binom{k+2}{2}.

(4.5)

Combining (4.4) and (4.5), we find that

\raise 0.21529pt\hbox{\small$\displaystyle\frac{|S_{k}|}{|S_{k+2}|}$}\leqslant\raise 0.21529pt\hbox{\small$\displaystyle\frac{(3+4C)^{2}(k+2)^{2}}{2(1+C)d_{\mathrm{min}}\bar{d}sn}$}.

Define $K=K(s):=\lfloor\frac{1}{2}(\alpha d_{\mathrm{min}}s-1)\rfloor$ , which implies by AM/GM that

\prod_{i=1}^{2K}\,(\lfloor\alpha d_{\mathrm{min}}s\rfloor+i)\leqslant\bigl{(}\lower 0.51663pt\hbox{\large$\textstyle\frac{3}{2}$}\alpha d_{\mathrm{min}}s\bigr{)}^{K}\quad\text{~{}for $\alpha d_{\mathrm{min}}s\geqslant 2$}.

This enables us to bound the probability that $k\leqslant\alpha d_{\mathrm{min}}s$ :

	$\displaystyle\raise 0.21529pt\hbox{\small$\displaystyle\frac{\sum_{k\leqslant\alpha d_{\mathrm{min}}s}\|S_{k}\|}{\sum_{j\geqslant 0}\|S_{j}\|}$}\leqslant\max_{k\leqslant\alpha d_{\mathrm{min}}s}\raise 0.21529pt\hbox{\small$\displaystyle\frac{\|S_{k}\|}{\|S_{k+2K}\|}$}$	$\displaystyle\leqslant\biggl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{(3+4C)^{2}}{2(1+C)d_{\mathrm{min}}\bar{d}sn}$}\biggr{)}^{\!K\,}\prod_{i=1}^{2K}\,(\lfloor\alpha d_{\mathrm{min}}s\rfloor+i)^{2}$
		$\displaystyle\leqslant\biggl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{9d_{\mathrm{min}}s}{32(1+C)\,\bar{d}n}$}\biggr{)}^{\!K}.$

Therefore, the probability that there is any set $S$ with $|S|=s$ having fewer than $\alpha d_{\mathrm{min}}s$ edges leaving is at most

P(s):=\binom{n}{s}\biggl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{9d_{\mathrm{min}}s}{32(1+C)\bar{d}n}$}\biggr{)}^{\!\alpha d_{\mathrm{min}}s/2-3/2}.

Using $(s+1)^{s+1}\leqslant 4s^{s+1}$ we find that $P(s+1)/P(s)$ is increasing with $s$ and is bounded by $\frac{81}{128}$ at $s=\frac{1}{2}n$ . Recalling that we can assume $s\geqslant(1-\alpha)d_{\mathrm{min}}$ , we conclude that

\operatorname{\mathbb{P}}\bigl{(}h(G)<\alpha d_{\mathrm{min}}\bigr{)}<3\,P((1-\alpha)d_{\mathrm{min}}).

It remains to simplify this bound. First we apply $\binom{n}{s}\leqslant\bigl{(}\frac{en}{s}\bigr{)}^{s}$ . Then we note that $\alpha\leqslant\frac{1}{6}$ and $\alpha d_{\mathrm{min}}\geqslant 4$ together imply that $\frac{1}{2}\alpha(1-\alpha)d_{\mathrm{min}}^{2}-(1-\alpha)d_{\mathrm{min}}-\frac{3}{2}\geqslant\frac{5}{26}\alpha d_{\mathrm{min}}^{2}$ . Then, under the same conditions,

3\,\biggl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{e}{1-\alpha}$}\biggr{)}^{(1-\alpha)d_{\mathrm{min}}}\biggl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{9(1-\alpha)}{32(1+C)}$}\biggr{)}^{\alpha(1-\alpha)d_{\mathrm{min}}^{2}/2-3/2}\leqslant 5_{\vphantom{x}}^{-\frac{5}{26}\alpha d_{\mathrm{min}}^{2}},

This completes the proof. ∎

4.2 Completing the proof of Theorem 2.8

Combining Corollary 2.5 and Theorem 4.3, we immediately get the following.

Theorem 4.4.

Let $G\sim\mathcal{G}(n,\boldsymbol{d})$ such that $d_{\mathrm{max}}\gg\log^{8}n$ and $d_{\mathrm{min}}\geqslant\gamma d_{\mathrm{max}}$ for some fixed constant $\gamma>0$ . Then, with probability at least $1-e^{-\Omega(d_{\mathrm{max}}^{2})}$ ,

\rho(G)=\widehat{\rho}(G)+O\mathopen{}\mathclose{{}\left(\lower 0.51663pt\hbox{\large$\textstyle\frac{\log^{2}d_{\mathrm{max}}}{d_{\mathrm{max}}}$}}\right).

Finally we show how to derive Theorem 2.8 from Theorem 4.2 and Theorem 4.4. If the degree sequence $\boldsymbol{d}$ satisfies (R1), that is, $d_{\mathrm{max}}^{2}=o(n)$ , then we have $d_{\mathrm{max}}^{2}=o(\bar{d}n)$ , since all the degrees are positive. Therefore, by Theorem 4.2, for fixed $\varepsilon>0$ ,

\operatorname{\mathbb{P}}\bigl{(}\rho(G)\geqslant\widehat{\rho}(G)+\varepsilon\bigr{)}\leqslant\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{{e^{n\varepsilon}-1}}$}n^{1/2}e^{O(d_{\mathrm{max}}^{2})}=e^{-O(n)}.

Assume that $d_{\mathrm{max}}^{2}=\Omega(n)$ and the degree sequence $\boldsymbol{d}$ satisfies (R2). Theorem 2.8 then follows from Theorem 4.4 by noting that $d_{\mathrm{max}}^{2}=\Omega(d_{\mathrm{max}}^{2}+n)$ and $(\log d_{\mathrm{max}})^{2}/d_{\mathrm{max}}=o(1)$ .

5 Numerical estimation via Eulerian partitions

The number of Eulerian orientations of a finite graph is a $\#P$ -complete problem equivalent to finding the permanent of a 0–1 matrix [29, 25]. However, the order of the matrix equals the number of edges in the graph, and the notorious difficulty of estimating large sparse permanents means that above about 100 edges we found it difficult to obtain accurate values.

Instead, we employed a repeatedly discovered theorem [18, Eq. (11)], [14], and also [5]. This result, stated below as Theorem 5.1, allowed us to obtain accurate estimates sometimes into thousand of vertices.

Let $G$ be a graph with even degrees $d_{1},\ldots,d_{n}$ . An Eulerian partition is a partition of the edges into undirected closed trails, where a trail is a walk that doesn’t repeat edges. Let $\mathcal{P}(G)$ denote the set of all Eulerian partitions, and note that

\lvert\mathcal{P}(G)\rvert=\prod_{i=1}^{n}\,\frac{d_{i}!}{(d_{i}/2)!\,2^{d_{i}/2}},

since each partition is uniquely described by a pairing of the edges at each vertex. For an Eulerian partition $P\in\mathcal{P}(G)$ , let $\lvert P\rvert$ denote the number of closed trails it comprises.

Theorem 5.1.

For any graph $G$ with even degrees $d_{1},\ldots,d_{n}$ ,

\rho(G)=\widehat{\rho}(G)+\raise 0.21529pt\hbox{\small$\displaystyle\frac{1}{n}$}\log T(G),~{}~{}\text{where}~{}~{}T(G):=\frac{1}{\lvert\mathcal{P}(G)\rvert}\sum_{P\in\mathcal{P}(G)}2^{\lvert P\rvert}.

Proof.

Say that an Eulerian partition $P$ and an Eulerian orientation $O$ are associated if each trail in $P$ is a directed closed walk in $O$ . In Figure 3 we give an example of an Eulerian orientation and an associated edge pairing.

Let $N$ be the number of pairs $(O,P)$ , where $O$ is an Eulerian orientation and $P$ is an associated Eulerian partition. Given $O$ , all the associated Eulerian partitions are obtained from a bijection at each vertex between the in-coming and out-going edges. That is,

N=\operatorname{EO}(G)\;\prod_{i=1}^{n}\,(d_{i}/2)!\,.

Conversely, given an Eulerian partition $P$ , the number of associated Eulerian orientations is clearly $2^{\lvert P\rvert}$ , so

N=\sum_{P\in\mathcal{P}(G)}2^{\lvert P\rvert}.

In view of the definitions (1.1) and (1.2), combining these counts gives the theorem. ∎

In Figure 4 we show the distribution of $\lvert P\rvert$ for two graphs with 256 vertices. $Q_{8}$ is the 8-dimensional hypercube (degree 8), while $C_{16}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{16}$ is the two-dimensional square lattice with periodic boundary conditions (degree 4).

To apply Theorem 5.1, we can generate many edge partitions at random and calculate the average $2^{\lvert P\rvert}$ . With careful programming, each trial requires about $20\,\lvert E(G)\rvert$ nanoseconds. Although $2^{\lvert P\rvert}$ is a highly skewed function, the average of a sequence of trials converges to a normal distribution as the number of trials increases. In most cases, we found 1000 averages based on at least one million trials each, then from those 1000 averages we found the 2-sigma confidence interval for the mean. The accuracy is better when $\lvert P\rvert$ is typically smaller, such as for higher degree or higher dimension.

6 Products of cycles

In this section we consider simple graphs which are Cartesian products of smaller graphs. After some general theory we will focus on products of cycles and their limits (infinite paths). The most famous example is “square ice” (the two-dimensional square lattice) which was the first periodic lattice whose residual entropy was determined exactly [17]. In Section 7 we will consider a different example that allows us to exemplify the accuracy of our estimates when the degree increases.

Suppose $G$ and $H$ are two graphs. The Cartesian product $G\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}H$ has vertex set $V(G)\times V(H)$ , with $(u,v)$ adjacent to $(u^{\prime},v^{\prime})$ if either $u=u^{\prime}$ and $v$ is adjacent to $v^{\prime}$ in $H$ , or $v=v^{\prime}$ and $u$ is adjacent to $u^{\prime}$ in $G$ .

In order to compare the residual entropy with our estimate $\rho_{\tau}$ defined in (2.5), we first consider the tree entropy. Let $G$ be a simple graph with $n$ vertices. Recall the definition of the Laplacian matrix $L(G)$ from Section 3.

The next simple lemma gives the spanning tree count of graph products in terms of eigenvalues of the Laplacian matrices.

Lemma 6.1.

Let $G$ and $H$ be two connected graphs on $\ell$ and $m$ vertices, respectively. Let $0=\mu_{0},\mu_{1},\ldots,\mu_{\ell-1}$ and $0=\mu^{\prime}_{0},\mu^{\prime}_{1},\ldots,\mu^{\prime}_{m-1}$ be the eigenvalues of the Laplacian matrices $L(G)$ and $L(H)$ , respectively. Then the number of spanning trees in the Cartesian product $G\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}H$ is

t(G\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}H)=\raise 0.21529pt\hbox{\small$\displaystyle\frac{1}{\ell m}$}\prod_{\begin{subarray}{c}0\leqslant j<\ell,\,0\leqslant k<m\\[0.90417pt] j+k\neq 0\end{subarray}}\!\!(\mu_{j}+\mu^{\prime}_{k}).

Proof.

By the Matrix Tree Theorem, $\ell m\,t(G\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}H)$ is equal to the product of the non-zero eigenvalues of $L(G\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}H)$ . In fact,

L(G\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}H)=I_{\ell}\otimes L(H)+L(G)\otimes I_{m},

where $\otimes$ is the matrix tensor product, from which it follows that the $\ell m$ eigenvalues of $L(G\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}H)$ are $\mu_{j}+\mu^{\prime}_{k}$ for $0\leqslant j<\ell$ and $0\leqslant k<m$ ; see for example [24]. ∎

6.1 Products of two cycles

Let $C_{m}$ denote the cycle with $m$ vertices. It is well known that the eigenvalues of $L(C_{m})$ are $2-2\cos\lower 0.51663pt\hbox{\large$\textstyle\frac{2\pi j}{m}$}$ for $0\leqslant j\leqslant m-1$ .

Lemma 6.2.

For $m\geqslant 3$ ,

	$\displaystyle\lim_{\ell\to\infty}\tau(C_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell})$	$\displaystyle=\raise 0.21529pt\hbox{\small$\displaystyle\frac{1}{m}$}\sum_{j=1}^{m-1}\log\,g\Bigl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{2\pi j}{m}$}\Bigr{)},\text{~{}where}$
	$\displaystyle g(y)$	$\displaystyle=2-\cos y+\sqrt{\cos^{2}y-4\cos y+3}.$

Also, $\lim_{\ell,m\to\infty}\tau(C_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell})\approx 1.1662436$ .

Proof.

By Lemma 6.1

\log t(C_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell})=-\log(\ell m)+\sum_{\begin{subarray}{c}0\leqslant j<\ell,\,0\leqslant k<m\\[0.90417pt] j+k\neq 0\end{subarray}}\log\Bigl{(}4-2\cos\lower 0.51663pt\hbox{\large$\textstyle\frac{2\pi j}{\ell}$}-2\cos\lower 0.51663pt\hbox{\large$\textstyle\frac{2\pi k}{m}$}\Bigr{)}.

As $\ell\to\infty$ , the sum over $j$ can be replaced by an integral, using

\int_{0}^{1}\log(1-a\cos(2\pi x))\,dx=\log\raise 0.21529pt\hbox{\small$\displaystyle\frac{1+\sqrt{1-a^{2}}}{2}$}\quad(\lvert a\rvert\leqslant 1).

This gives the first claim after some elementary manipulation. The second claim comes from integrating $\log g(y)$ and is exactly $\textstyle\frac{4}{\pi}$ times Catalan’s constant [6]. ∎

The value of $\lim_{m\to\infty}\rho(C_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{m})$ was famously determined by Lieb [17] in 1967 to be exactly $\log\lower 0.51663pt\hbox{\large$\textstyle\frac{8\sqrt{3}}{9}$}\approx 0.43152$ . This compares poorly to Pauling’s estimate $0.40547$ but very well to our estimate $\lim_{m\to\infty}\rho_{\tau}(C_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{m})\approx 0.43054$ .

$G$	$\tau(G)$	$\rho(G)$	$\rho_{\tau}(G)$
$C_{3}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.04453	0.46210	0.49140
$C_{4}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.09917	0.46299	0.46408
$C_{5}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.12373	0.44216	0.45180
$C_{6}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.13687	0.44577	0.44523
$C_{7}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.14472	0.43690	0.44130
$C_{8}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.14979	0.43960	0.43877
$C_{9}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.15326	0.43477	0.43703
$C_{10}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.15574	0.43672	0.43579
$C_{11}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.15757	0.43369	0.43488
$C_{12}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.15895	0.43514	0.43419
$C_{13}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.16003	0.43308	0.43365
$C_{14}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.16089	0.43418	0.43322
$C_{15}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.16158	0.43269	0.43287
$C_{16}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.16215	0.43356	0.43259

Table 1: Parameters for

C_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell}

\ell\to\infty

To further illustrate the usefulness of our estimate, we considered the case where the square lattice is finite in one direction; i.e., a square lattice on an infinitely long cylinder. For $3\leqslant m\leqslant 14$ , we obtained precise values of $\lim_{\ell\to\infty}\rho(C_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell})$ using the transfer matrix method which we describe in the following theorem. These values and the corresponding estimates $\rho_{\tau}$ are presented in Table 1 and Figure 5. It is seen that $\rho_{\tau}$ tracks $\rho$ very well as $m$ increases, especially for large $m$ .

Theorem 6.3.

Let $G$ be an Eulerian graph with vertices $\{1,\ldots,n\}$ . For $\boldsymbol{z}=(z_{1},\ldots,z_{n})\in\mathbb{Z}^{n}$ , define $N_{G}(\boldsymbol{z})$ to be the number of orientations of $G$ such that $d_{\mathrm{out}}(v)-d_{\mathrm{in}}(v)=z_{v}$ for $1\leqslant v\leqslant n$ , where $d_{\mathrm{in}}(v),d_{\mathrm{out}}(v)$ are the in-degree and out-degree of vertex $v$ . Define the $2^{n}\times 2^{n}$ matrix $T=(t_{\boldsymbol{x},\boldsymbol{y}})$ , whose rows and columns are indexed by $\boldsymbol{x},\boldsymbol{y}\in\{0,1\}^{n}$ , by $t_{\boldsymbol{x},\boldsymbol{y}}=N_{G}(2(\boldsymbol{y}-\boldsymbol{x}))$ . Then

\lim_{\ell\to\infty}\rho(G\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell})=\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{n}$}\log\lambda,

where $\lambda$ is the largest eigenvalue of $T$ . Furthermore, let $\varGamma$ be the group action on $\{0,1\}^{n}$ induced by the automorphism group of $G$ acting on the coordinate positions, together with the involution $\boldsymbol{x}\mapsto(1,\ldots,1)-\boldsymbol{x}$ . Suppose this action has orbits $O_{1},\ldots,O_{m}$ . Define the $m\times m$ matrix $S=(s_{i,j})$ where $s_{i,j}$ is the common row sum of the submatrix of $T$ induced by rows $O_{i}$ and columns $O_{j}$ . Then $S$ has the same largest eigenvalue $\lambda$ .

Proof.

The rationale for $T$ was described by Lieb [17] and we will be content with sketching a proof of the last part. Note that, by symmetry and converse, $t_{\boldsymbol{x}^{\gamma},\boldsymbol{y}^{\gamma}}=t_{\boldsymbol{x},\boldsymbol{y}}$ for $\boldsymbol{x},\boldsymbol{y}\in\{0,1\}^{n}$ and $\gamma\in\varGamma$ . This implies that if $\boldsymbol{v}$ is a positive eigenvector of $T$ with eigenvalue $\lambda$ , then so is $\boldsymbol{v}^{\gamma}$ . By averaging over $\gamma\in\varGamma$ , we find a non-zero eigenvector of $T$ corresponding to eigenvalue $\lambda$ which takes a constant value $r_{i}$ on each orbit $O_{i}$ . Then $(r_{1},\ldots,r_{n})$ is an eigenvector of $S$ with eigenvalue $\lambda$ . Conversely, any eigenvector of $S$ becomes an eigenvector of $T$ with the same eigenvalue on replicating its value in each orbit. ∎

The advantage of the matrix $S$ is that it can be much smaller than $T$ . For example, the final row in Table 2 below has $T$ matrix of order 1,048,576 but its $S$ matrix has order 7,456.

6.2 Products of three cycles

Define $R_{m}=C_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{m}$ to be a finite simple cubic lattice with periodic boundary conditions. We did not find an estimate of the residual entropy of this lattice in the literature. From (1.3) and (2.1) we have

0.91623\leqslant\lim_{m\to\infty}\rho(R_{m})\leqslant 1.0925.

From [28] we have $\lim_{m\to\infty}\tau(R_{m})\approx 1.67338$ , so our estimate is $\lim_{m\to\infty}\rho_{\tau}(R_{m})=0.9251$ .

To judge the accuracy of our estimate, we computed values of $\rho(R_{m})$ up to $m=20$ using the method of Section 5. Precise values become difficult to obtain past $n=16$ (4096 vertices). In Figure 6 we compare $\rho(R_{m})$ to $\rho_{\tau}(R_{m})$ and again observe good correlation between $\rho_{\tau}$ and $\rho$ . Using rational extrapolation of $\rho(R_{m})$ , we believe that

\lim_{m\to\infty}\rho(R_{m})=0.9252\pm 0.0002.

In other words, we cannot distinguish between $\lim_{m\to\infty}\rho(R_{m})$ and $\lim_{m\to\infty}\rho_{\tau}(R_{m})$ .

$G$	$\tau(G)$	$\rho(G)$	$\rho_{\tau}(G)$
$C_{3}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{3}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.61344	0.95055	0.95511
$C_{3}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{4}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.63332	0.94486	0.94521
$C_{3}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{5}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.64164	0.93930	0.94101
$C_{3}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{6}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.64605	0.93857	0.93881
$C_{4}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{4}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.64941	0.93703	0.93713
$C_{4}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{5}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	1.65593	0.93382	0.93387

Table 2: Parameters for

\lim_{n\to\infty}C_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{n}

We also show in Table 2 results for the Cartesian product of three cycles where two of the cycles have small fixed length. We computed $\tau(G)$ using a simple extension of Lemma 6.2, and $\rho(G)$ using the transfer matrix method. Note that $\widehat{\rho}(G)=0.91629$ in all cases.

7 Cycles of cliques

In this section, we find the residual entropy of cycles of cliques which are products $K_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell}$ . Our reason for studying this family is that increasing $m$ allows us to test Conjecture 2.2 and to observe how $\rho_{\tau}$ and $\rho$ are related as the degree increases. Throughout this section, we assume that $m$ is odd since the number of Eulerian orientations is zero otherwise.

7.1 Residual entropy

Recall that $\operatorname{RT}(m)=\operatorname{EO}(K_{m})$ is the number of regular tournaments.

Theorem 7.1.

If $n=m\ell\to\infty$ and $m$ is odd then

\rho(K_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell})=\raise 0.21529pt\hbox{\small$\displaystyle\frac{1}{m}$}\log\biggl{(}\operatorname{RT}(m+2)\Bigm{/}\binom{m+1}{(m{+}1)/2}\biggr{)}+O\Bigl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{\log m}{m\ell}$}\Bigr{)}.

We start the proof with a sequence of auxiliary lemmas.

Lemma 7.2.

Let $\boldsymbol{f}\in\{-2,0,2\}^{m}$ be such that $\sum_{i\in[m]}f_{i}=0$ . Let $\operatorname{NT}(m,\boldsymbol{f})$ denote the number of tournaments with $m$ vertices that $\boldsymbol{f}$ is the vector of differences of out-degrees and in-degrees. Then,

\operatorname{NT}(m,\boldsymbol{f})=\raise 0.21529pt\hbox{\small$\displaystyle\frac{2}{\binom{m+1}{(m+1)/2}^{2}}$}\,e^{c(\boldsymbol{f})}\operatorname{RT}(m+2),

where $|c(\boldsymbol{f})|\leqslant 2+O(m^{-1})$ uniformly over such $\boldsymbol{f}$ as $m\to\infty$ .

Proof.

From [23, Theorem 4.4], we find that

\operatorname{NT}(m,\boldsymbol{f})=\operatorname{RT}(m)\exp\biggl{(}-\raise 0.21529pt\hbox{\small$\displaystyle\frac{\sum_{i\in[m]}f_{i}^{2}}{2m}$}+O(m^{-1})\biggr{)}.

(7.1)

Note that $\lower 0.51663pt\hbox{\large$\textstyle\frac{\sum_{i\in[m]}f_{i}^{2}}{2m}$}\in[0,2]$ .

Next, consider a tournament $T$ with vertices $V\cup\{u,v\}$ , where $\lvert V\rvert=m$ . Let $T^{\prime}$ be the subtournament of $T$ induced by $V$ . There are $\binom{m+1}{(m+1)/2}^{2}/2$ ways to choose the neighbours of $u$ and $v$ so that their in-degrees and out-degrees are equal. For each of those cases, a particular $\boldsymbol{f}(T^{\prime})\in\{-2,0,2\}^{m}$ is necessary and sufficient for $T$ to be regular. That is, $\operatorname{RT}(m+2)$ is the sum of $\binom{m+1}{(m+1)/2}^{2}/2$ terms, each having value given by (7.1). This completes the proof. ∎

Now we consider a cycle of cliques $K_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell}$ . Take a clockwise cyclic ordering of the cliques $K_{m}^{1},\ldots,K_{m}^{\ell}$ . Given any orientation $D$ of the cycle of cliques, we define the net flow $f_{(i-1)\to i}(D)$ as the difference of the number of clockwise arcs and anticlockwise arcs between two consecutive cliques $K_{m}^{i-1}$ and $K_{m}^{i}$ in $D$ for all $i-1,i$ modulo $\ell$ .

Lemma 7.3.

If $D$ is an Eulerian orientation of $K_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell}$ then, for all $i,j\in[\ell]$ ,

f_{(i-1)\to i}(D)=f_{(j-1)\to j}(D).

Proof.

It suffices to show that $f_{(i-1)\to i}(D)=f_{i\to(i+1)}(D)$ for all $i\in[\ell]$ . Using that $D$ is Eulerian, we observe that

f_{i\to(i+1)}(D)-f_{(i-1)\to i}(D)=\sum_{v\in K_{m}^{i}}d^{+}(v)-\sum_{v\in K_{m}^{i}}d^{-}(v)=0,

where $d^{+}$ and $d^{-}$ denote the out-degree and in-degree of a vertex in $D$ , respectively. ∎

We introduce a class $A_{m,f}$ of digraphs, for $m,f$ odd and $|f|\leqslant m$ . These digraphs correspond to an Eulerian orientation of $K_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell}$ induced on the clique $K_{m}^{i}$ and with previous and subsequent cliques replaced by two vertices $s$ and $t$ . Such graphs have $m+2$ vertices including these two special vertices. There is no directed edge between $s$ and $t$ , but exactly one directed edge between each other pair. Vertices $v\neq s,t$ have $d^{-}(v)=d^{+}(v)$ , while $d^{+}(s)=d^{-}(t)=\lower 0.51663pt\hbox{\large$\textstyle\frac{m+f}{2}$}$ and $d^{-}(s)=d^{+}(t)=\lower 0.51663pt\hbox{\large$\textstyle\frac{m-f}{2}$}$ . Next, we study the rates of decrease of the quantity $a_{m,f}:=\lvert A_{m,f}\rvert$ decreases with respect to $f$ , using the following lemma.

Lemma 7.4.

Suppose a bipartite graph $G$ with two parts $X,Y$ has no isolated vertices. Suppose that there is a constant $c$ such that $d(x)\leqslant cd(y)$ for every edge $xy$ with $x\in X,y\in Y$ . Then $|Y|\leqslant c\,|X|$ .

Proof.

We have

|Y|=\sum_{xy\in G}\raise 0.21529pt\hbox{\small$\displaystyle\frac{1}{d(y)}$}\leqslant c\sum_{xy\in G}\raise 0.21529pt\hbox{\small$\displaystyle\frac{1}{d(x)}$}=c\,|X|

as required. ∎

Lemma 7.5.

For odd $m,f$ with $|f|\leqslant m$ ,
(a) $a_{m,-f}=a_{m,f}$ .
(b) If $f\geqslant 3$ then

\raise 0.21529pt\hbox{\small$\displaystyle\frac{a_{m,f}}{\binom{m}{(m+f)/2}}$}\leqslant\raise 0.21529pt\hbox{\small$\displaystyle\frac{a_{m,f-2}}{\binom{m}{(m+f-2)/2}}$}\leqslant\raise 0.21529pt\hbox{\small$\displaystyle\frac{a_{m,1}}{\binom{m}{(m+1)/2}}$}=\raise 0.21529pt\hbox{\small$\displaystyle\frac{\operatorname{RT}(m+2)}{\binom{m+1}{(m+1)/2}}$}.

Proof.

Part (a) is proved by interchanging the roles of $s$ and $t$ .

We next prove the first inequality of part (b). Define a bipartite graph $G$ with parts $A_{m,f}$ and $A_{m,f-2}$ . For $x\in A_{m,f}$ and $y\in A_{m,f-2}$ , $x$ is adjacent to $y$ if $y$ is obtained by reversing a directed path of length 2 from $s$ to $t$ . Consider such an edge $xy$ . The vertex set $V(x)\setminus\{s,t\}$ can be partitioned into four parts: $V_{1}$ is adjacent from $s$ and to $t$ ; $V_{2}$ , with $\lower 0.51663pt\hbox{\large$\textstyle\frac{m+f}{2}$}-\lvert V_{1}\rvert$ vertices, is adjacent from both $s$ and $t$ ; $V_{3}$ , with $\lower 0.51663pt\hbox{\large$\textstyle\frac{m+f}{2}$}-\lvert V_{1}\rvert$ vertices, is adjacent to both $s$ and $t$ ; and finally $V_{4}$ , with $\lvert V_{1}\rvert-f$ vertices, is adjacent to $s$ and from $t$ . Note that $\lvert V_{1}\rvert=\lvert V_{4}\rvert+f>0$ so $G$ has no isolated vertices in the first part, and the same argument with the parts interchanged shows that there are no isolated vertices in the second part either.

Directed paths of length 2 from $s$ to $t$ correspond to vertices in $V_{1}$ . Reversing one path takes us to $y$ , where there are $\lvert V_{4}\rvert+1$ directed paths from $t$ to $s$ . Thus, $d(x)=\lvert V_{1}\rvert$ and $d(y)=\lvert V_{4}\rvert+1=\lvert V_{1}\rvert-f+1$ .

Lemma 7.4 now gives us

\raise 0.21529pt\hbox{\small$\displaystyle\frac{a_{m,f}}{a_{m,f-2}}$}\leqslant\max_{A_{m,f}}\raise 0.21529pt\hbox{\small$\displaystyle\frac{\lvert V_{1}\rvert-f+1}{\lvert V_{1}\rvert}$}.

This expression is increasing with $\lvert V_{1}\rvert$ , which is at most $\textstyle\frac{m+f}{2}$ . Observe that the ratio of the corresponding binomials is also $\frac{m-f+2}{m+f}$ . This establishes the claimed monotonicity of $\frac{a_{m,f}}{\binom{m}{(m+f)/2}}$ with respect to $f$ .

To establish claim we observe that any digraph from $A_{m,1}$ corresponds to a regular tournament on $m+2$ vertices with a specified direction between $s$ and $t$ . Thus,

\raise 0.21529pt\hbox{\small$\displaystyle\frac{a_{m,1}}{\binom{m}{(m+1)/2}}$}=\raise 0.21529pt\hbox{\small$\displaystyle\frac{\operatorname{RT}(m+2)}{2\binom{m}{(m+1)/2}}$}=\raise 0.21529pt\hbox{\small$\displaystyle\frac{\operatorname{RT}(m+2)}{\binom{m+1}{(m+1)/2}}$},

as required. ∎

Now we are ready to establish the residual entropy of a cycle of cliques.

Proof of Theorem 7.1.

Let $\operatorname{EO}_{f}(m,\ell)$ denote the number of Eulerian orientations of $K_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell}$ with net flow $f\in[-m,m]$ . Note that from Lemma 7.3 we know that the flow between any two consecutive cliques is the same. Therefore,

\operatorname{EO}(K_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell})=\sum_{\begin{subarray}{c}|f|\leqslant m\\ \text{$f$ is odd}\end{subarray}}\operatorname{EO}_{f}(m,\ell).

(7.2)

Any orientation with flow $f$ can be represented as a sequence of $\ell-1$ choices of $D^{i}\in A_{m,f}$ , $i=1,\ldots,\ell-1$ , and then a choice of orientations of the edges of the last clique $K_{m}^{\ell}$ with specified differences of out and in degrees from $\{-2,0,2\}$ . Note that the orientations of edges of $t$ -vertices of $D^{i}\in A_{m,f}$ should match the orientations of the $s$ -vertices of $D^{i+1}$ so we need to adjust the count by a factor $\binom{m}{(m-f)/2}$ for the choices of $D^{i+1}$ given $D^{i}$ . Therefore,

\operatorname{EO}_{f}(m,\ell)=\biggl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{a_{m,f}^{\ell-1}}{\binom{m}{(m{-}f)/2}}$}\biggr{)}^{\ell-1}\binom{m}{(m{-}f)/2}N_{f}(m,\ell),

where $N_{f}(m,\ell)$ is an average number of choices for the orientations of the edges of the last clique $K_{m}^{\ell}$ given the orientations of all other edges of $K_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell}$ . Using Lemma 7.2 and Lemma 7.5, we obtain

\operatorname{EO}_{f}(m,\ell)\leqslant\operatorname{EO}_{1}(m,\ell)e^{2+O(m^{-1})}=\biggl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{\operatorname{RT}(m+2)}{\binom{m+1}{(m+1)/2}}$}\biggr{)}^{\ell}e^{O(1)}.

Substitution this bound into (7.2), we find that

\biggl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{\operatorname{RT}(m+2)}{\binom{m+1}{(m+1)/2}}$}\biggr{)}^{\ell}e^{O(1)}\leqslant\operatorname{EO}(K_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell})\leqslant m\biggl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{\operatorname{RT}(m+2)}{\binom{m+1}{(m+1)/2}}$}\biggr{)}^{\ell}e^{O(1)}.

Taking the logarithm and dividing by the number of vertices $n=m\ell$ completes the proof. ∎

7.2 Pauling’s estimate and the tree-entropy correction

A cycle of cliques $K_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell}$ has degree $m+1$ . Therefore, Pauling’s estimate (1.2) is

\widehat{\rho}(K_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell})=\log\binom{m+1}{(m+1)/2}-\raise 0.21529pt\hbox{\small$\displaystyle\frac{m+1}{2}$}\log 2.

In Table 3 we compare the exact residual entropy given by Theorem 7.1 with Pauling’s estimate (1.2) and our estimate (2.5) for $m$ up to $35$ . The exact values of $\operatorname{RT}(m+2)$ are taken from [9, Table 1]. Recall that our estimate $\rho_{\tau}$ requires the spanning tree entropy, which is given by the next lemma.

$G$	degree	$\tau(G)$	$\rho(G)$	$\widehat{\rho}(G)$	$\rho_{\tau}(G)$
$K_{3}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	4	1.04453	0.46210	0.40547	0.49140
$K_{5}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	6	1.53988	0.97656	0.91629	0.99189
$K_{7}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	8	1.87255	1.53422	1.47591	1.54351
$K_{9}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	10	2.12402	2.11892	2.06369	2.12514
$K_{11}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	12	2.32634	2.72190	2.66983	2.72635
$K_{13}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	14	2.49561	3.33800	3.28887	3.34134
$K_{15}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	16	2.64109	3.96395	3.91748	3.96655
$K_{17}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	18	2.76862	4.59755	4.55347	4.59963
$K_{19}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	20	2.88213	5.23725	5.19532	5.23896
$K_{21}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	22	2.98438	5.88195	5.84195	5.88337
$K_{23}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	24	3.07739	6.53079	6.49253	6.53199
$K_{25}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	26	3.16268	7.18313	7.14646	7.18416
$K_{27}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	28	3.24143	7.83847	7.80324	7.83936
$K_{29}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	30	3.31457	8.49640	8.46249	8.49718
$K_{31}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	32	3.38283	9.15658	9.12388	9.15727
$K_{33}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	34	3.44682	9.81876	9.78718	9.81937
$K_{35}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\infty}$	36	3.50704	10.48270	10.45215	10.48325

Table 3: Residual entropy

\lim_{\ell\to\infty}\rho(K_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell})

compared to its estimates

\widehat{\rho}

and

\rho_{\tau}

Lemma 7.6.

For $\ell,m\geqslant 3$ , the number of spanning trees in $K_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell}$ is

t(K_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell})=\raise 0.21529pt\hbox{\small$\displaystyle\frac{\ell}{m}$}\Bigl{(}\Bigl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{\sqrt{m}+\sqrt{m+4}}{2}$}\Bigl{)}^{\!2\ell}+\Bigl{(}\raise 0.21529pt\hbox{\small$\displaystyle\frac{\sqrt{m}+\sqrt{m+4}}{2}$}\Bigl{)}^{\!-2\ell\,}\Bigr{)}^{m-1}.

Moreover,

\tau(K_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell})=\begin{cases}\lower 0.51663pt\hbox{\large$\textstyle\frac{\log(\ell/m)}{\ell m}$}+\lower 0.51663pt\hbox{\large$\textstyle\frac{2(m-1)}{m}$}\log\lower 0.51663pt\hbox{\large$\textstyle\frac{\sqrt{m}+\sqrt{m+4}}{2}$}+O(\ell^{-1}m^{-2\ell}),&\text{as \,$\ell m\to\infty$;}\\[4.30554pt] \lower 0.51663pt\hbox{\large$\textstyle\frac{\log(\ell/m)}{\ell m}$}+\log m-\lower 0.51663pt\hbox{\large$\textstyle\frac{\log m-2}{m}$}+O(m^{-2}),&\text{as \,$m\to\infty$.}\end{cases}

Proof.

The non-zero eigenvalues of $L(C_{\ell})$ are $2-2\cos\bigl{(}\frac{2\pi j}{\ell}\bigr{)}$ for $1\leqslant j\leqslant\ell-1$ , while the non-zero eigenvalues of $L(K_{m})$ are all equal to $m$ . Therefore, we have from Lemma 6.1 that

	$\displaystyle t(K_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell})$	$\displaystyle=\ell m^{m-2}\prod_{j=1}^{\ell-1}\,\Bigl{(}m+2-2\cos\lower 0.51663pt\hbox{\large$\textstyle\frac{2\pi j}{\ell}$}\Bigr{)}^{m-1}$
		$\displaystyle=\ell m^{-1}T(m,\ell)^{m-1},$

where

T(m,\ell)=\prod_{j=0}^{\ell-1}\,\Bigl{(}m+2-2\cos\lower 0.51663pt\hbox{\large$\textstyle\frac{2\pi j}{\ell}$}\Bigr{)}.

The latter product can be recognised as $2(-1)^{\ell}T_{2\ell}(im^{1/2}/2)-2$ , where $T_{2\ell}(x)$ is the Chebyshev polynomial in its standard normalization. The lemma now follows from the explicit form of $T_{2\ell}$ . ∎

Finally, for large $m$ , we show that Pauling’s estimate (1.2) approximates the exact residual entropy up to an error $O\bigl{(}\frac{\log m}{m}\bigr{)}$ , thus confirming Conjecture 2.2 for cycles of cliques. Our new heuristic $\rho_{\tau}$ has a much smaller error, again demonstrating its remarkable precision.

Theorem 7.7.

If $m$ is odd and $m\to\infty$ then

	$\displaystyle\rho(K_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell})$	$\displaystyle=\lower 0.51663pt\hbox{\large$\textstyle\frac{m+2}{2}$}\log 2-\lower 0.51663pt\hbox{\large$\textstyle\frac{m-1}{2m}$}\log m-\lower 0.51663pt\hbox{\large$\textstyle\frac{\log\pi}{2}$}-\lower 0.51663pt\hbox{\large$\textstyle\frac{3}{2m}$}+O\mathopen{}\mathclose{{}\left(\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{m^{2}}$}+\lower 0.51663pt\hbox{\large$\textstyle\frac{\log m}{m\ell}$}}\right)$
		$\displaystyle=\widehat{\rho}(K_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell})+\lower 0.51663pt\hbox{\large$\textstyle\frac{\log m}{2m}$}-\lower 0.51663pt\hbox{\large$\textstyle\frac{3}{4m}$}+O\mathopen{}\mathclose{{}\left(\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{m^{2}}$}+\lower 0.51663pt\hbox{\large$\textstyle\frac{\log m}{m\ell}$}}\right)$
		$\displaystyle=\rho_{\tau}(K_{m}\mathbin{\raise 0.6458pt\hbox{$\scriptstyle\square$}}C_{\ell})+O\mathopen{}\mathclose{{}\left(\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{m^{2}}$}+\lower 0.51663pt\hbox{\large$\textstyle\frac{\log m}{m\ell}$}}\right),$

where $\widehat{\rho}$ and $\rho_{\tau}$ are defined in (1.2) and (2.5).

Proof.

The value of $\operatorname{RT}(m+2)$ follows from (2.3). We also have that

\binom{m+1}{(m+1)/2}=\biggl{(}1-\raise 0.21529pt\hbox{\small$\displaystyle\frac{1}{4m}$}+O(m^{-2})\biggr{)}\raise 0.21529pt\hbox{\small$\displaystyle\frac{2^{m+1}}{\sqrt{\pi(m+1)/2}}$}.

Using Theorem 7.1 and routine asymptotic expansions, we obtain the first claim. The other two equalities follow from (1.2), (2.5) and the second part of Lemma 7.6. ∎

8 Other examples

8.1 Triangular lattice $T_{n}$ and Baxter’s constant

For the triangular lattice $T_{n}$ of degree 6 on $n$ vertices, with periodic boundary conditions, Baxter [1] proved in 1969 that

\lim_{n\to\infty}\rho(T_{n})=\log\lower 0.51663pt\hbox{\large$\textstyle\frac{3\sqrt{3}}{2}$}\approx 0.95477.

This compares poorly with Pauling’s estimate $\widehat{\rho}(T_{n})=\log\frac{5}{2}\approx 0.91629$ . From [6], we know that the spanning tree entropy is

\lim_{n\to\infty}\tau(T_{n})=\raise 0.21529pt\hbox{\small$\displaystyle\frac{5}{\pi}$}\sum_{i\geqslant 1}\raise 0.21529pt\hbox{\small$\displaystyle\frac{\sin(i\pi/3)}{i^{2}}$}\approx 1.61530,

so our estimate (2.5) gives

\rho_{\tau}(T_{n})=\widehat{\rho}(T_{n})+\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{2}$}\tau_{6}-\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{2}$}\tau(T_{n})\approx 0.95417,

which is very close to the correct value.

8.2 3-dimensional ice

Of the several regular structures of water ice, we consider hexagonal ice (Ih) and cubic ice (Ic). Using a heuristic series expansion, Nagle [26] judged both $\rho(\mathrm{Ih})$ and $\rho(\mathrm{Ic})$ to lie in the interval [0.40992,0.41012]. By extrapolating finite simulations to the limit, Kolafa [10] obtained the slightly higher value $\rho\approx 0.41043$ for both types of ice. Neither Nagle’s nor Kolafa’s calculations are sufficient to positively distinguish $\rho(\mathrm{Ih})$ from $\rho(\mathrm{Ic})$ .

Using the method of Lyons [19] we find that the spanning tree entropy of ice Ic is

	$\displaystyle\tau(\mathrm{Ic})$	$\displaystyle=\raise 0.21529pt\hbox{\small$\displaystyle\frac{1}{8}$}\int_{0}^{1}\!\!\int_{0}^{1}\!\!\int_{0}^{1}\log\bigl{(}16464-3136(c_{x}+c_{y}+c_{z})-2016(c_{x}c_{y}+c_{x}c_{z}+c_{y}c_{z})$
		$\displaystyle\kern 90.00014pt-960c_{x}c_{y}c_{z}+16(c_{x}^{2}c_{y}^{2}+c_{x}^{2}c_{z}^{2}+c_{y}^{2}c_{z}^{2})$
		$\displaystyle\kern 90.00014pt-32(c_{x}^{2}c_{y}c_{z}+c_{x}c_{y}^{2}c_{z}+c_{x}c_{y}c_{z}^{2})\bigr{)}\,dx\,dy\,dz$
		$\displaystyle\approx 1.20645995,$

where $c_{u}$ means $\cos(2\pi u)$ . Nagle [26] noticed that the generating function for walks returning to the origin is the same for both types of ice, implying that the eigenvalue distributions are the same and thus $\tau(\mathrm{Ih})=\tau(\mathrm{Ic})$ , which we verified to high precision. Consequently, we have

\rho_{\tau}(\mathrm{Ih})=\rho_{\tau}(\mathrm{Ic})\approx 0.410433,

in excellent agreement with Kolafa’s estimate.

8.3 Hypercubes $Q_{d}$

The number of Eulerian orientations of a $d$ -dimensional hypercube $Q_{d}$ on $n=2^{d}$ vertices is only known up to $d=6$ [30, sequence A358177], and it appears that even the asymptotic value is unknown.

$d$	$n$	$\widehat{\rho}$	$\rho_{\tau}$	$\rho$	$\widehat{\rho}-\rho$	$\rho_{\tau}-\rho$
4	16	0.405465	0.464780	0.499770	-0.0943	-0.035
6	64	0.916291	0.948381	0.955050	-0.0388	-0.0067
8	256	1.475907	1.489316	1.490759	-0.0149	-0.0014
10	1024	2.063693	2.069225	2.069554	-0.0059	-0.00033
12	4096	2.669829	2.672343	2.672420	-0.0026	-0.00008
14	16384	3.288868	3.290206	3.290224	-0.0014	-0.00002

Table 4: Parameters for hypercubes

Q_{d}

Using the method described in Section 5, we have computed estimates of $\rho(Q_{d})$ up to $d=14$ . From [3], we know that

t(Q_{d})=\lower 0.51663pt\hbox{\large$\textstyle\frac{1}{n}$}\prod_{i=1}^{d}\,(2i)^{\binom{d}{i}}.

The values shown for $\rho(Q_{d})$ in Table 4 are believed correct to within one value of the final digit. It is seen that Pauling’s estimate $\widehat{\rho}(Q_{d})$ is improving as the dimension increases, but our estimate $\rho_{\tau}(Q_{d})$ is approaching the right answer more quickly. Experimentally, it seems likely that $\rho(Q_{d})=\rho_{\tau}(Q_{d})+O(2^{-d})$ .

9 Concluding remarks

We conclude with a short summary of interesting problems on the residual entropy of graphs mentioned in this paper that remain open.

(a)

Prove Conjecture 2.2.
(b)

Give a combinatorial explanation for the strong correlation between residual entropy and spanning tree entropy. Theorem 2.4 gives an analytic explanation for denser graphs. A qualitative hint is that the presence of short cycles tends to reduce the spanning tree count [20] (at least for sparse graphs) but, by Theorem 5.1, tends to increase the number of Eulerian orientations.
(c)

Determine whether $\rho(\mathrm{Ih})$ and $\rho(\mathrm{Ic})$ coincide.
(d)

Find the asymptotics of $\rho(Q_{d})$ for hypercubes.
(e)

Investigate the analogue of $\rho_{\tau}(G)$ for irregular graphs. The average number of spanning trees for a wide range of degree sequences was determined in [7].

References

[1] R. J. Baxter, F model on a triangular lattice, J. Math. Physics, 10 (1969) 1211–1216.
[2] F. Bencs, M. Borbényi and P. Csikvári, Number of Eulerian orientations for Benjamini–Schramm convergent graph sequences, preprint arXiv:2409.18012, 2024.
[3] O. Bernardi, On the spanning trees of the hypercube and other products of graphs, Electron. J. Combinatorics, 19 (2012) 4, #P51.
[4] B. Bollobás, The isoperimetric number of random regular graphs, Europ. J. Combin., 9 (1988) 241–244.
[5] B. Bollobás, Evaluations of the circuit partition polynomial, J. Combin. Th., Ser. B, 85 (2002) 261-268.
[6] M. L. Glasser and F. Y. Wu, On the entropy of spanning trees on a large triangular lattice, Ramanujan J., 10 (2005) 205–214.
[7] C. Greenhill, M. Isaev, M. Kwan and B. D. McKay, The average number of spanning trees in sparse graphs with given degrees, Europ. J. Combin., 63 (2017) 6–25.
[8] M. Isaev, T. Iyer and B. D. McKay, Asymptotic enumeration of orientations of a graph as a function of the out-degree sequence, Electron. J. Combin., 27 (2020) #P1.26.
[9] M. Isaev, B. D. McKay and R.-R. Zhang, Cumulant expansion for counting Eulerian orientations, J. Combin. Th., Ser. B, 172 (2025) 263–314.
[10] J. Kolafa, Residual entropy of ices and clathrates from Monte Carlo simulation, J. Chem. Phys., 140 (2014) #204507.
[11] B. Kolesnik and N. Wormald, Lower bounds for the isoperimetric numbers of random regular graphs, SIAM J. Disc. Math., 28 (2014) 553–575.
[12] A. V. Kostochka, The number of spanning trees in graphs with a given degree sequence, Random Structures Algorithms, 6 (1995) 269–274.
[13] M. Krivelevich, B. Sudakov, V. Vu and N. C. Wormald, Random regular graphs of high degree, Random Structures Algorithms, 18 (2001) 346–363.
[14] M. Las Vergnas, Le polynôme de Martin d’un graphe eulérien, in Combinatorial mathematics (Marseille–Luminy, 1981), Ann. Discr. Math. 17, North Holland, 1983, 397–411.
[15] M. Las Vergnas, An upper bound for the number of Eulerian orientations of a regular graph, Combinatorica, 10 (1990) 61–65.
[16] D-Z. Li, W-J. Huang, Y. Yao and X-B. Yang, Exact results for the residual entropy of ice hexagonal monolayer, Phys. Rev. E, 107 (2023) #054121.
[17] E. H. Lieb, Residual entropy of square ice, Physical Review, 162 (1967) 162–172.
[18] E. H. Lieb and F. Y. Wu, Two-dimensional ferroelectric models, in Phase Transitions and Critical Phenomena, C. Domb and M. Green eds., vol. 1, Academic Press, 1972, 331–490.
[19] R. Lyons, Asymptotic enumeration of spanning trees, Combinatorics Probability Computing, 14 (2005) 491–522.
[20] B. D. McKay, Spanning trees in regular graphs, Europ. J. Combin., 4 (1983) 149–160.
[21] B. D. McKay, Asymptotics for 0-1 matrices with prescribed line sums, in Enumeration and Design, Academic Press, 1984, 225–238.
[22] B. D. McKay, Asymptotics for symmetric 0-1 matrices with prescribed row sums, Ars Combinatoria, 19A (1985) 15–26.
[23] B. D. McKay and X. Wang, Asymptotic enumeration of tournaments with a given score sequence, J. Combin. Th., Ser. A, 73 (1996) 77–90.
[24] R. Merris, Laplacian graph eigenvectors, Linear Algebra Appl., 278 (1998) 221–236.
[25] M. Mihail, P. Winkler, On the number of Eulerian orientations of a graph, Algorithmica, 16 (1996), 402–414.
[26] J. F. Nagle, Lattice statistics of hydrogen bonded crystals. I. The residual entropy of ice, J. Math. Phys., 7 (1966) 1484–1491.
[27] L. Pauling, The structure and entropy of ice and of other crystals with some randomness of atomic arrangement, J. Amer. Chem. Soc., 57 (1935) 2680–2684.
[28] A. Rosengren, On the number of spanning trees for the 3D simple cubic lattice, J. Phys. A: Math. Gen., 20 (1987) #L923.
[29] A. Schrijver, Bounds on the number of Eulerian orientations, Combinatorica, 3 (1983) 375–380.
[30] N. J. A. Sloane et al., The Online Encyclopedia of Integer Sequences, https://oeis.org.

Correlation between residual entropy and spanning tree entropy of ice-type models on graphs

Abstract

1 Introduction

2 Statements of the main results and conjectures

Conjecture 2.1 ([29]).

Conjecture 2.2.

Lemma 2.3.

2.1 Sufficiently growing degrees and good expansion

Theorem 2.4 (Isaev, McKay, Zhang [9]).

Corollary 2.5.

2.2 A new upper bound on EO⁡(G)\operatorname{EO}(G)

Theorem 2.6.

Corollary 2.7.

2.3 Random graphs with given degrees

Theorem 2.8.

2.4 A new heuristic estimate for regular graphs

Theorem 2.9.

Proof.

3 Proof of Theorem 2.6, Corollary 2.5 and Lemma 2.3

Proof of Theorem 2.6..

Lemma 3.1 (Kostochka [12]).

Proof.

Proof of Corollary 2.7.

Proof of Lemma 2.3.

Proof of Corollary 2.5.

4 Proof of Theorem 2.8

Lemma 4.1.

Proof.

Theorem 4.2.

Proof.

4.1 Expansion properties of random graphs with given degrees

Theorem 4.3.

Proof.

4.2 Completing the proof of Theorem 2.8

Theorem 4.4.

5 Numerical estimation via Eulerian partitions

Theorem 5.1.

Proof.

6 Products of cycles

Lemma 6.1.

Proof.

6.1 Products of two cycles

Lemma 6.2.

Proof.

Theorem 6.3.

Proof.

6.2 Products of three cycles

7 Cycles of cliques

7.1 Residual entropy

Theorem 7.1.

Lemma 7.2.

Proof.

Lemma 7.3.

Proof.

Lemma 7.4.

Proof.

Lemma 7.5.

Proof.

Proof of Theorem 7.1.

7.2 Pauling’s estimate and the tree-entropy correction

Lemma 7.6.

Proof.

Theorem 7.7.

Proof.

8 Other examples

8.1 Triangular lattice TnT_{n} and Baxter’s constant

8.2 3-dimensional ice

8.3 Hypercubes QdQ_{d}

9 Concluding remarks

References

2.2 A new upper bound on $\operatorname{EO}(G)$

8.1 Triangular lattice $T_{n}$ and Baxter’s constant

8.3 Hypercubes $Q_{d}$