The first eigenvector of a Distance Matrix
is nearly constant

We start by explaining the first problem which first arose implicitly in [20]: let $G=(V,E)$ be a finite, connected graph on $n$ vertices $V=\left\{v_{1},\dots,v_{n}\right\}$ and let $D\in\mathbb{R}^{n\times n}$ denote the graph distance matrix defined by ${D}_{ij}=d(v_{i},v_{j})$ where $d(v_{i},v_{j})$ is the graph distance between the vertices $v_{i},v_{j}$.

Problem 1. The linear system of equations

$$Dx=\mathbb{1}=(1,1,\dots,1)\in\mathbb{R}^{n}$$

tends to have a solution $x\in\mathbb{R}^{n}$ for ‘most’ graphs. Why is that?

It is certainly the case that for ‘many’ graphs, the matrix $D$ simply happens to be invertible and this case is not particularly interesting. However, when the graph $G$ exhibits additional structure and/or symmetries, then this is often reflected in the non-invertibility of $D$, however, even then the linear system seems to have a solution for a constant right-hand side. We can make this quantitative: the database of graphs in Mathematica 12 contains 7642 connected graphs with $3\leq|V|\leq 50$ vertices. Among those, the matrix $D$ is not invertible for 3155 of these graphs (this large percentage is due to the fact that the database contains more ‘structured’ than ‘generic’ graphs). However, $Dx=\mathbb{1}$ has a solution in all but 5 cases.
Using the Monte Carlo method, we estimate that for Erdős-Renyi graphs $G(n,p)$ the likelihood of a linear system associated to $G(50,p)$ not having a solution is (uniformly in $p$) less than $1\%$ and decays rapidly for larger $n$. The phenomenon arose in [20] where solutions of $Dx=\mathbb{1}$ are used to construct a notion of curvature on graphs. Naturally, the effectiveness of such a notion hinges on whether a solution of $Dx=\mathbb{1}$ typically exists. Our starting point is a spectral existence result.

The right-hand side is positive but need not be large. Note that $D$ vanishes on the diagonal and thus the sum of the eigenvalues is 0. Given our assumption about the sign of the eigenvalues, this implies $\lambda_{1}=|\lambda_{2}|+\dots+|\lambda_{n}|$. We would thus expect that, generically, $\lambda_{1}\gg|\lambda_{2}|$ which makes the right-hand side rather small. Hence, the applicability of Proposition 1 depends on whether $\left\langle v,\mathbb{1}/\sqrt{n}\right\rangle$ is close to 1, whether the first eigenvector $v$ is ‘nearly constant’.

We note that, by Cauchy-Schwarz, all $v\in\mathbb{R}^{n}_{\geq 0}$ of unit length $\|v\|=1$ satisfy

with equality if and only if all the entries of $v$ are $1/\sqrt{n}$.

The Theorem can be interpreted as saying that the Cauchy-Schwarz inequality is close to being attained (which happens if the vectors $v$ and $\mathbb{1}$ are linearly dependent, meaning that $v$ is close to a constant vector). The two inequalities in the Theorem are sharp: if we consider the star graph, then the minimum of the largest eigenvector is attained in the center and behaves asymptotically as described in the Theorem. For the second inequality we consider a metric space where $n-1$ points are distance $\leq\varepsilon$ (for suitable $\varepsilon$ depending on $n$) from each other with an additional point at distance $1$ from the first $n-1$ points. We refer to §2.3 for details.

A search in the Mathematica database identifies the ‘sun’ graphs (also sometimes known as trampoline graph, see [9]) as having the smallest constant within the library. A computation shows that the limiting behavior of the constant for sun graphs is given by $(1/2+1/\sqrt{5})^{1/2}\sim 0.973$.

Motivated by the extremal metric space (see Fig. 1), it is natural to consider so-called broom graphs or comets [14, 23]. They are obtained by taking a star and attaching a path to the central vertex. Numerics suggest that they might lead to a relatively small inner product $\left\langle v,\mathbb{1}/\sqrt{n}\right\rangle$ (certainly quite a bit smaller than the sun graphs and possibly even close to $1/\sqrt{2}$), however, their leading eigenvector does not seem to be known. We conclude by formally stating the second problem.

Problem 2. If $G=(V,E)$ is an unweighted, connected graph on $n$ vertices and $v$ is the leading eigenvector of its distance matrix $D$, then it seem for most graphs $\left|\left\langle v,\mathbb{1}\right\rangle\right|\sim(1-\varepsilon)\sqrt{n}$ with a very small value of $\varepsilon$. Can this be made precise and/or quantified?

Let $(X,d)$ be a metric space and $x_{1},\dots,x_{n}\in X$. The largest eigenvector of the distance matrix then describes the global maximum of the energy functional $E:\mathbb{R}^{n}\rightarrow\mathbb{R}$ given by

This is naturally related to the study of energy integrals of the form

where $\mu$ is a probability measure. The study of such measures is classical, we refer to the 1958 paper of Björck [6] as well as Alexander [1, 2], Alexander & Stolarsky [3], Gross [13], Wolf [24, 25] and references therein. Both in the continuous [6] as well as in the discrete case [21] the optimal measure $\mu$ tends to be localized in a rather small set contained in the boundary. We observe that by switching from probability measures (essentially $L^{1}$) to $L^{2}$, the geometry of the problem changes substantially: having mass localized is actually rather expensive and it is beneficial to spread things out. Our main result makes this precise. The distance matrix for graphs was introduced by Graham & Pollak [12]. Papendieck & Recht [16] studied the largest possible entry of the principal eigenvector. Das [10, 11] considered the problem of bounding the extremal entries of the principal eigenvectors of the distance matrix. The survey of Aouchiche & Hansen [4] lists several more results about the principal eigenvalue and principal eigenvector, we also refer to the books of Stanić [19], Stevanović [22]. Distance matrices in the Euclidean setting were introduced by Schoenberg [18] (see also Young & Householder [26]) and studied in [5, 7, 8, 26]. We caution that the classical Euclidean Distance Matrix (Menger [15]) is different and has $D_{ij}=\|x_{i}-x_{j}\|^{2}$.

Remark. The proof suggests that the only way for the argument to be close to sharp is if the triangle inequality is close to being an equation for most cases. It seems interesting, albeit nontrivial, to see whether one can obtain quantitative improvements in settings where this is not the case.