On the inverse of the adjacency matrix of a graph

Abstract

A real symmetric matrix G with zero diagonal encodes the adjacencies of the vertices of a graph G with weighted edges and no loops. A graph associated with a n × n non-singular matrix with zero entries on the diagonal such that all its (n - 1) × (n - 1) principal submatrices are singular is said to be a NSSD. We show that the class of NSSDs is closed under taking the inverse of G. We present results on the nullities of one- and two-vertex deleted subgraphs of a NSSD. It is shown that a necessary and sufficient condition for two-vertex deleted subgraphs of G and of the graph Λ(G-1) associated with G-1 to remain NSSDs is that the submatrices belonging to them, derived from G and G-1, are inverses. Moreover, an algorithm yielding what we term plain NSSDs is presented. This algorithm can be used to determine if a graph G with a terminal vertex is not a NSSD.