Inverse of the adjacency matrices and strong anti-reciprocal eigenvalue property

Abstract

Let G be a graph having a unique perfect matching M, $A\left(G\right)$ be the adjacency matrix of G and $W\left(G\right)$ be the collection of all positive weight functions defined on the edge set of G in which each weight function w assigns weight 1 to each matching edge and a positive weight to each non-matching edge. The weighted graph $G_w$ satisfies the $(-SR)$ property if for each eigenvalue of $G_w$, its anti-reciprocal is also an eigenvalue of $G_w$ with the same multiplicity. In this paper, a class of graphs with a unique perfect matching M for which the diagonal entries of the inverse of the adjacency matrix of each graph are all zero is investigated. Furthermore, it is shown that no noncorona graph in this class satisfies the $(-SR)$ property even for a single weight function $w\in W\left(G\right)$.

Keywords:

• Weighted graph
• perfect matching
• alternating path
• strong anti-reciprocal eigenvalue property
• corona graph

2010 Mathematics Subject Classifications:

• 05C50
• 15A18

Disclosure statement

No potential conflict of interest was reported by the author(s).