Class of weighted graphs with strong anti-reciprocal eigenvalue property. Issue 6 (2nd June 2020)

• Record Type: Journal Article
• Authors: Ahmad, Uzma; Hameed, Saira; Jabeen, Shaista
• Is Part Of: Linear & multilinear algebra. Volume 68:Issue 6(2020)
• Page Start: 1129
• Page End: 1139
• Publication Date: 2020-06-02
• Subjects: Sukanta Pati; Weighted graph -- perfect matching -- alternating path -- spectrum -- strong anti-reciprocal property -- corona graph; 05C50 -- 15A18; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2018.1532489
• Languages: English
• ISSNs: 0308-1087

Abstract

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.