Characterizing graphs with fully positive semidefinite $Q$-matrices

For $q\in \mathbb{R}$, the $Q$-matrix $Q=Q_q$ of a connected simple graph $G=(V,E)$ is $Q_q=(q^{\mathrm{\partial }(x,y)}{)}_{x,y\in V}$, where $\mathrm{\partial }$ 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 $K_{2,3}$.