Characterizing graphs with fully positive semidefinite Q-matrices

Abstract

For q∈R, the Q-matrixQ=Qq of a connected simple graph G=(V,E) is Qq=(q∂(x,y))x,y∈V, where ∂ denotes the path-length distance. Describing the set π(G) consisting of those q∈R for which Qq 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 π(G) is easily seen to be a nonempty closed subset of the interval [−1,1]. In this note, we show that π(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 K2,3.