On non-bipartite graphs with strong reciprocal eigenvalue property

Abstract

Let G be a simple connected graph and A(G) be the adjacency matrix of G. A diagonal matrix with diagonal entries ±1 is called a signature matrix. If A(G) is nonsingular and X=SA(G)−1S−1 is entrywise nonnegative for some signature matrix S, then X can be viewed as the adjacency matrix of a unique weighted graph. It is called the inverse of G, denoted by G+. A graph G is said to have the reciprocal eigenvalue property (property(R)) if A(G) is nonsingular, and 1λ is an eigenvalue of A(G) whenever λ is an eigenvalue of A(G). Further, if λ and 1λ have the same multiplicity for each eigenvalue λ, then G is said to have the strong reciprocal eigenvalue property (property (SR)). It is known that for a tree T, the following conditions are equivalent: a) T+ is isomorphic to T, b) T has property (R), c) T has property (SR) and d) T is a corona tree (it is a tree which is obtained from another tree by adding a new pendant at each vertex).Studies on the inverses, property (R) and property (SR) of bipartite graphs are available in the literature. However, their studies for the non-bipartite graphs are rarely done. In this article, we study the inverse and property (SR) for non-bipartite graphs. We first introduce an operation, which helps us to study the inverses of non-bipartite graphs. As a consequence, we supply a class of non-bipartite graphs for which the inverse graph G+ exists and G+ is isomorphic to G. It follows that each graph G in this class has property (SR).