On the inverse of a class of weighted graphs

Abstract

In this article, only connected bipartite graphs $G$ with a unique perfect matching $\c{M}$ are considered. Let $G_\w$ denote the weighted graph obtained from $G$ by giving weights to its edges using the positive weight function $\w:E(G)\ar (0,\ity)$ such that $\w(e)=1$ for each $e\in\c{M}$. An unweighted graph $G$ 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 $1$ or $-1$. 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 $G$ with a unique perfect matching such that $G$ is invertible. That class is denoted by $\c{H}_{nmc}$. It is natural to ask whether $G_\w$ is invertible for each invertible graph $G\in\c{H}_{nmc}$ 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 $G\in\c{H}_{nmc}$ and a weight function$\w\not\equiv\1$ such that $G_\w$ is not invertible. Then the weight functions $\w$ for each graph $G\in\c{H}_{nmc}$ such that $G_\w$ is invertible, are characterized.