Oriented spanning trees and stationary distribution of digraphs

Let $G$ be a weighted digraph with vertex set $V(G)$ and edge set $E(G)$, and each edge $e=ij\in E(G)$ is weighted by an indeterminate $w_{e}(G)=w_{ij}(G)$. The notation $e=ij\in E(G)$ means there exists a directed edge from tail vertex $i$ to head vertex $j$, and the tail and head of $e$ are denoted by $i=t(e)$ and $j=h(e)$, respectively. The outdegree and the indegree of a vertex $v$ in $G$ are denoted by $d_{v}^{+}(G)$ and $d_{v}^{-}(G)$, respectively. The word “weighted” is omitted when $w_{e}(G)=1$ for each $e\in E(G)$. The line digraph $\mathcal{L}(G)$ of a digraph $G$ has vertex set $V(\mathcal{L}(G))=E(G)$, and there exits directed edge from $e$ to $f$ in $\mathcal{L}(G)$ if and only if $h(e)=t(f)$.

An oriented spanning tree [8, 14] of weighted digraph $G$ is a subtree containing all vertices of $G$, in which one vertex, the root, has outdegree $0$, and every other vertex has outdegree $1$. Let $\mathbb{T}_{u}(G)$ denote the set of oriented spanning trees of $G$ with root $u$, and let $\kappa_{u}(G)=|\mathbb{T}_{u}(G)|$. A spanning tree enumerator of $G$ is defined as

If $w_{e}(G)=1$ for each $e\in E(G)$, then $t_{u}(G)=\kappa_{u}(G)$ is the number of oriented spanning trees of $G$ with root $u$. Some algebraic formulas for $\kappa_{u}(G)$ and $t_{u}(G)$ were given in [7, 8, 16].

In the study of random walks on digraphs, one basic problem is to determine the stationary distribution vector, which represents the long-term behaviour of the Markov chain associated with the digraph [2]. The stationary distribution has applications in PageRank algorithms [4].

There exists a closed relation between the stationary distribution vector and spanning tree enumerators of digraphs. For a strongly connected digraph $G$ with $n$ vertices, its stationary distribution vector $\pi=(\pi_{1},\ldots,\pi_{n})^{\top}$ satisfies [3, 12] $\pi_{i}=\frac{t_{i}(\widetilde{G})}{\sum_{j=1}^{n}t_{j}(\widetilde{G})}$, where $\widetilde{G}$ is the weighted digraph obtained from $G$ by taking $w_{ij}(\widetilde{G})=d_{i}^{+}(G)^{-1}$ for each $ij\in E(G)$. Let $m_{ij}$ be the mean first passage time from $i$ to $j$, then the value $\mathcal{K}(G)=\sum_{j\in V(G),j\neq i}m_{ij}\pi_{j}$ (does not depend on $i$) is called the Kemeny’s constant [12] of $G$.

Some formulas for counting spanning trees in undirected line graphs can be found in [9, 10, 18, 19]. For line digraphs, Knuth [13] proved the following formula for counting oriented spanning trees. Combinatorial and bijective proofs of the Knuth formula are given in [15] and [6], respectively.

Let $\mathcal{E}(G)$ denote the number of Eulerian circuits of a digraph $G$. By Lemma 2.6, we can derive the following formula for counting Eulerian circuits of a line digraph from Theorem 1.1.

Let $\{w_{i}(G)\}_{i\in V(G)}$ be indeterminates on $V(G)$. If each edge $\{u,v\}$ in $G$ has weight $w_{uv}(G)=w_{v}(G)$, then we say that the weights of $G$ are induced by $\{w_{i}(G)\}_{i\in V(G)}$.

Levine [14] proved the following formula for spanning tree enumerators of a line digraph, which is a generalization of Theorem 1.1.

A biclique [11] is a bipartite digraph $Q$ whose vertices can be partitioned into two parts $Q^{(1)}$ and $Q^{(2)}$, and $E(Q)=\{ij:i\in Q^{(1)},j\in Q^{(2)}\}$. A biclique partition of a digraph $G$ is a set $\varepsilon=\{Q_{1},\ldots,Q_{r}\}$ of bicliques in $G$ such that each edge of $G$ belongs to exactly one biclique of $\varepsilon$. For a biclique partition $\varepsilon=\{Q_{1},\ldots,Q_{r}\}$ of $G$, its biclique digraph has vertex set $\varepsilon=\{Q_{1},\ldots,Q_{r}\}$ and edge set $\{Q_{i}Q_{j}:Q_{i}^{(2)}\cap Q_{j}^{(1)}\neq\emptyset\}$.

It is known that every digraph $G$ has a biclique partition $\varepsilon$ satisfying $|\varepsilon|\leq|V(G)|$ ($|\varepsilon|$ is much smaller than $|V(G)|$ for many digraphs). So we can get reduction formulas for spanning tree enumerators, stationary distribution vector and Kemeny’s constant of $G$ by counting oriented spanning trees in the biclique digraph of $\varepsilon$. Moreover, based on Observation 1.4, it is natural to use biclique partitions to extend the results of Knuth [13] and Levine [14] from line digraphs to general digraphs.

In Section 2, we give some basic definitions, notations and auxiliary lemmas. In Section 3, we give biclique partition formulas for counting oriented spanning trees and Eulerian circuits of digraphs, which generalize Theorems 1.1-1.3 to general digraphs. In Section 4, we give biclique partition formulas for stationary distribution vector and Kemeny’s constant of digraphs. In Section 5, we give some concluding remarks, including more general spanning tree identity in digraphs, and the method for enumerating spanning trees of undirected graphs by vertex degrees and biclique partitions.

Let $G$ be a weighted digraph on $n$ vertices, and each edge $e=ij\in E(G)$ is weighted by an indeterminate $w_{ij}(G)$. The weighted degree of a vertex $i$ is $d_{i}(G)=\sum_{ij\in E(G)}w_{ij}(G)$. The Laplacian matrix $L_{G}$ is the $n\times n$ matrix with entries

Let $A(i,j)$ denote the submatrix of a matrix $A$ obtained by deleting the $i$-th row and the $j$-th column, and let $\det(A)$ denote the determinant of a square matrix $A$. The following lemma follows from the all minors matrix tree theorem [7].

The following is a fomula for counting Eulerian circuits of digraphs.

Let $G$ be a strongly connected weighted digraph with $n$ vertices, and all weights of $G$ are positive. The transition probability matrix $P_{G}$ of $G$ is the $n\times n$ matrix with entries

A random walk on $G$ is defined by $P_{G}$, that is, $(P_{G})_{ij}$ denotes the probability of moving from vertex $i$ to vertex $j$. Notice that $P_{G}$ is an irreducible nonnegative matrix with spectral radius $1$, and the all-ones vector is a right eigenvector for the eigenvalue $1$. By the Perron-Frobenius theorem, there exists a unique positive vector $\pi(G)=(\pi_{1}(G),\ldots,\pi_{n}(G))^{\top}$ such that $\pi(G)^{\top}P_{G}=\pi(G)^{\top}$ and $\sum_{i=1}^{n}\pi_{i}(G)=1$. Such vector $\pi(G)$ is called the stationary distribution vector [5] of $G$.

Let $m_{ij}$ be the mean first passage time from vertex $i$ to vertex $j$, then the value $\mathcal{K}(G)=\sum_{j\in V(G),j\neq i}m_{ij}\pi_{j}(G)$ (does not depend on $i$) is called the Kemeny’s constant of $G$.

For a matrix $E$, let $E[i_{1},\ldots,i_{s}|j_{1},\ldots,j_{t}]$ denote an $s\times t$ submatrix of $E$ whose row indices and column indices are $i_{1},\ldots,i_{s}$ and $j_{1},\ldots,j_{t}$, respectively. The following is a determinant identity involving the Schur complement.

A biclique is a bipartite digraph $Q$ whose vertices can be partitioned into two parts $Q^{(1)}$ and $Q^{(2)}$, and $E(Q)=\{ij:i\in Q^{(1)},j\in Q^{(2)}\}$. Let $\varepsilon=\{Q_{1},\ldots,Q_{r}\}$ be a biclique partition of a weighted digraph $G$ whose weights are induced by indeterminates $\{w_{i}(G)\}_{i\in V(G)}$. Let $\Omega(\varepsilon)$ denote the weighted biclique digraph with vertex set $\varepsilon=\{Q_{1},\ldots,Q_{r}\}$ and edge set $\{Q_{i}Q_{j}:Q_{i}^{(2)}\cap Q_{j}^{(1)}\neq\emptyset\}$, and the weight of $Q_{i}Q_{j}$ in $\Omega(\varepsilon)$ is

where $w(Q_{j})=\sum_{u\in Q_{j}^{(2)}}w_{u}(G)$. By Observation 1.4, we know that the part (1) of the following theorem extends Theorem 1.3 to general digraphs.

We can deduce the following result in [14] from part (2) of Theorem 3.10.

For a biclique partition $\varepsilon=\{Q_{1},\ldots,Q_{r}\}$ of digraph $G$, let $\Theta(\varepsilon)$ denote the weighted biclique digraph with vertex set $\varepsilon=\{Q_{1},\ldots,Q_{r}\}$ and edge set $\{Q_{i}Q_{j}:Q_{i}^{(2)}\cap Q_{j}^{(1)}\neq\emptyset\}$, and the weight of $Q_{i}Q_{j}$ in $\Theta(\varepsilon)$ is

Clearly, $\Theta(\varepsilon)$ is obtained from $\Omega(\varepsilon)$ by taking $w_{v}(G)=1$ for each $v\in V(G)$ in equation (3.1). We can deduce the following result from Theorem 3.10.

By Observation 1.4, we know that the following formulas extend Theorem 1.2 to general Eulerian digraphs.

Let $\pi(G)$ denote the stationary distribution vector of a digraph $G$. By using biclique partitions of digraphs, we give the following reduction formulas for stationary distribution vector and Kemeny’s constant of digraphs.