On the inverse of a class of weighted graphs

In this article, only connected bipartite graphs G with a unique perfect matching M are considered. Let Gw denote the weighted graph obtained from G by giving weights to its edges using the positive weight function w : E(G)→ (0,∞) such that w(e) = 1 for each e ∈ M. An unweighted graph G may be viewed as a weighted graph with the weight function w ≡ 1 (all ones vector). A weighted graph Gw is nonsingular if its adjacency matrix A(Gw) is nonsingular. The inverse of a nonsingular weighted graph Gw is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A(Gw) 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. 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 Hnmc. It is natural to ask whether Gw is invertible for each invertible graph G ∈ Hnmc and for each weight function w 6≡ 1. In this article, first an example is given to show that there is an invertible graph G ∈ Hnmc and a weight function w 6≡ 1 such that Gw is not invertible. Then the weight functions w for each graph G ∈ Hnmc such that Gw is invertible, are characterized.