On non-bipartite graphs with strong reciprocal eigenvalue property

Abstract

Let G be a simple connected graph and $A\left(G\right)$ be the adjacency matrix of G. A diagonal matrix with diagonal entries ±1 is called a signature matrix. If $A\left(G\right)$ is nonsingular and $X=SA{\left(G\right)}^{-1}{S}^{-1}$ 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 $A\left(G\right)$ is nonsingular, and $\frac{1}{\lambda }$ is an eigenvalue of $A\left(G\right)$ whenever λ is an eigenvalue of $A\left(G\right)$. Further, if λ and $\frac{1}{\lambda }$ 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).