Class of weighted graphs with strong anti-reciprocal eigenvalue property. Issue 6 (2nd June 2020) - Ahmad, Uzma; Hameed, Saira; Jabeen, Shaista
Abstract : Let G be a graph with a unique perfect matching and W ( G ) 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 w ∈ W ( G ), let G w denote the weighted graph obtained from G corresponding to the weight function w . Let A ( G w ) denote the adjacency matrix of G w . We say G w satisfies the anti-reciprocal eigenvalue property (property (−R)) if for each eigenvalue λ of A ( G w ), its anti-reciprocal − 1 / λ is also an eigenvalue of A ( G w ) . Furthermore, if λ and − 1 / λ have the same multiplicities, then we say G w 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 G w with a unique perfect matching satisfies property (−SR) for each w ∈ W ( G ) if and only if it is a corona graph.