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Let  G  be a graph having a unique perfect matching  M ,be the adjacency matrix of  G  andbe 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 graphsatisfies theproperty if for each eigenvalue of, its anti-reciprocal is also an eigenvalue ofwith 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 theproperty even for a single weight function.

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