On The Inverse Of A Class Of Bipartite Graphs With Unique Perfect Matchings

Abstract

Let G be a simple, undirected graph and Gw be the weighted graph obtained from G by giving weights to its edges using a positive weight function w. A weighted graph Gw is said to be nonsingular if its adjacency matrix A(Gw) is nonsingular. In [9], Godsil has given a class $\mathcal{G }$of connected, unweighted, bipartite, nonsingular graphs G with a unique perfect matching, such that A(G)â1 is signature similar to a nonnegative matrix, that is, there exists a diagonal matrix D with diagonal entries ±1 such that DA(G)â1D is nonnegative. The graph associated to the matrix DA(G)â1D is called the inverse of G and it is denoted by G+. The graph G+ is an undirected, weighted, connected, bipartite graph with a unique perfect matching. Nonsingular, unweighted trees are contained inside the class G. We first give a constructive characterization of the class of weighted graphs Hw that can occur as the inverse of some graph Gâ\mathcal{ G}. This generalizes Theorem 2.6 of Neumann and Pati[13], where the authors have characterized graphs that occur as inverses of nonsingular, unweighted trees. A weighted graph Gw is said to have the property (R) if for each eigenvalue λ of A(Gw), 1âλ is also an eigenvalue of A(Gw). If further, the multiplicity of λ and 1âλ are the same, then Gw is said to have property (SR). A characterization of the class of nonsingular, weighted trees Tw with at least 8 vertices that have property (R) was given in [13] under some restriction on the weights. It is natural to ask for such a characterization for the whole of G, possibly with some weaker restrictions on the weights. We supply such a characterization. In particular, for trees it settles an open problem raised in [13].