Characterizing graphs with fully positive semidefinite $Q$-matrices

TL;DR

This paper characterizes graphs for which the $Q$-matrix is positive semidefinite for all real $q$ in [-1,1], linking this property to hypercube embeddability and specific graph configurations.

Contribution

It provides a complete characterization of graphs with $Q_q$ positive semidefinite for all $q$ in [-1,1], connecting spectral properties to geometric and combinatorial graph features.

Findings

$ ext{pi}(G) = [-1,1]$ iff $G$ is hypercube embeddable

Graphs with $ ext{pi}(G) = [-1,1]$ are bipartite and exclude certain five-vertex configurations

The characterization links spectral properties to geometric embedding and forbidden subgraphs.

Abstract

For $q\in\mathbb{R}$, the $Q$-matrix $Q=Q_q$ of a connected simple graph $G=(V,E)$ is $Q_q=(q^{\partial(x,y)})_{x,y\in V}$, where $\partial$ denotes the path-length distance. Describing the set $\pi(G)$ consisting of those $q\in \mathbb{R}$ for which $Q_q$ is positive semidefinite is fundamental in asymptotic spectral analysis of graphs from the viewpoint of quantum probability theory. Assume that $G$ has at least two vertices. Then $\pi(G)$ is easily seen to be a nonempty closed subset of the interval $[-1,1]$. In this note, we show that $\pi(G)=[-1,1]$ if and only if $G$ is isometrically embeddable into a hypercube (infinite-dimensional if $G$ is infinite) if and only if $G$ is bipartite and does not possess certain five-vertex configurations, an example of which is an induced $K_{2,3}$.