Class of weighted graphs with strong anti-reciprocal eigenvalue property

Abstract

Let G be a graph with a unique perfect matching and be the collection of all positive weight functions defined on the edge set of G such that each weight function assigns weight 1 to each matching edge and a positive weight to each non-matching edge of G. Given , let denote the weighted graph obtained from G corresponding to the weight function w. Let denote the adjacency matrix of . We say satisfies the anti-reciprocal eigenvalue property (property (−R)) if for each eigenvalue λ of , its anti-reciprocal is also an eigenvalue of . Furthermore, if λ and have the same multiplicities, then we say satisfies the strong anti-reciprocal eigenvalue property (property (−SR)). In this paper, the graphs with property (−SR) in the class of graphs with unique perfect matching are investigated and it is shown that a weighted graph with a unique perfect matching satisfies property (−SR) for each if and only if it is a corona graph.