Graph Invertibility

TL;DR

This paper explores the concept of graph invertibility, providing criteria for bipartite graphs with perfect matchings, extending to multigraphs, and offering methods to construct large families of invertible graphs.

Contribution

It introduces necessary and sufficient conditions for graph invertibility, generalizes the concept to multigraphs, and presents a procedure for constructing new invertible graphs.

Findings

Characterization of invertible bipartite graphs with perfect matchings

Criteria for invertibility of multigraphs

A method to construct large families of invertible graphs

Abstract

Extending the work of Godsil and others, we investigate the notion of the inverse of a graph (specifically, of bipartite graphs with a unique perfect matching). We provide a concise necessary and sufficient condition for the invertibility of such graphs and generalize the notion of invertibility to multigraphs. We examine the question of whether there exists a "litmus subgraph" whose bipartiteness determines invertibility. As an application of our invertibility criteria, we quickly describe all invertible unicyclic graphs. Finally, we describe a general combinatorial procedure for iteratively constructing invertible graphs, giving rise to large new families of such graphs.