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On the inverse of a class of weighted graphs
No full textIn this article, only connected bipartite graphs with a unique perfect matching are considered. Let G_\w denote the weighted graph obtained from by giving weights to its edges using the positive weight function \w:E(G)\ar (0,\ity) such that \w(e)=1 for each . An unweighted graph may be viewed as a weighted graph with the weight function \w\equiv\1 (all ones vector). A weighted graph G_\w is nonsingular if its adjacency matrix A(G_\w) is nonsingular. The {\em inverse} of a nonsingular weighted graph G_\w is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A(G_\w) via a diagonal matrix whose diagonal entries are either or . In [S.K.~Panda and S.~Pati. On some graphs which possess inverses. {\em Linear and Multilinear Algebra}, 64:1445--1459, 2016.], the authors characterized a class of bipartite graphs with a unique perfect matching such that is invertible. That class is denoted by . It is natural to ask whether G_\w is invertible for each invertible graph and for each weight function \w\not\equiv\1. In this article, first an example is given to show that there is an invertible graph and a weight function\w\not\equiv\1 such that G_\w is not invertible. Then the weight functions \w for each graph such that G_\w is invertible, are characterized
