Invertibility of graphs with a unique perfect matching

TL;DR

This paper explores the invertibility properties of graphs with a unique perfect matching, introducing negative invertibility and characterizing such graphs for small sizes, revealing unique properties like self-invertibility.

Contribution

It extends the concept of graph invertibility to negatively invertible graphs and characterizes all small graphs with a unique perfect matching regarding their invertibility.

Findings

Negatively invertible graphs exhibit self-invertibility.

Complete characterization of graphs with a unique perfect matching for m ≤ 6.

Identification of properties distinguishing positive and negative invertibility.

Abstract

In this paper we investigate invertibility of graphs with a unique perfect matching, i.e. graphs having a unique 1-factor. We recall the new notion of the so-called negatively invertible graphs investigated by the authors in the recent paper. It is an extension of the classical definition of an inverse graph due to Godsil. We characterize all graphs with a unique perfect matching on $m\le 6$ vertices with respect to their positive and negative invertibility. We show that negatively invertible graphs exhibit properties like selfinvertibility which cannot be observed within the class of positively invertible non-bipartite graphs with a unique perfect matching.