Inverse Graphs of Non-Singular Bicyclic Graphs without a Unique Perfect Matching

The inverse of the adjacency matrix of a graph has been extensively studied in algebraic graph theory. Several results characterize inverse graphs for graphs with unique perfect matchings. However, much less is known about graphs without unique perfect matchings. In this paper, we consider the class of bicyclic graphs that do not possess a unique perfect matching. Among all such graphs, we identify those that are non-singular. We characterize the non-singular bicyclic graphs in this class whose adjacency matrices have inverses with zero diagonal entries. We further investigate the structure of the graph associated with the inverse of the adjacency matrix and prove that there exists a unique bicyclic graph in this class whose associated inverse graph is also bicyclic.