Bicyclic Graphs with Unicyclic or Bicyclic Inverses

Abstract

A graph $G$ is nonsingular if its adjacency matrix $A(G)$ is nonsingular. A nonsingular graph $G$ is said to have an inverse $G^+$ if $A(G)^{−1}$ is signature similar to a nonnegative matrix. Let $\mathcal{H}$ be the class of connected bipartite graphs with unique perfect matchings. We present a characterization of bicyclic graphs in $\mathcal{H}$ which possess unicyclic or bicyclic inverses.