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