On a Construction of Integrally Invertible Graphs and their Spectral Properties

TL;DR

This paper introduces new methods for constructing integrally invertible graphs, explores their spectral properties, and provides a classification of small graphs with unique 1-factors regarding their invertibility.

Contribution

It proposes novel bridging operations for constructing integrally invertible graphs and analyzes their spectral characteristics, including bounds on eigenvalues.

Findings

Established sufficient conditions for positive and negative invertibility.

Derived lower bounds for the least positive eigenvalue of constructed graphs.

Provided a census of small graphs with unique 1-factors and their invertibility properties.

Abstract

Godsil (1985) defined a graph to be invertible if it has a non-singular adjacency matrix whose inverse is diagonally similar to a nonnegative integral matrix; the graph defined by the last matrix is then the inverse of the original graph. In this paper we call such graphs positively invertible and introduce a new concept of a negatively invertible graph by replacing the adjective `nonnegative' by `nonpositive in Godsil's definition; the graph defined by the negative of the resulting matrix is then the negative inverse of the original graph. We propose new constructions of integrally invertible graphs (those with non-singular adjacency matrix whose inverse is integral) based on an operation of `bridging' a pair of integrally invertible graphs over subsets of their vertices, with sufficient conditions for their positive and negative invertibility. We also analyze spectral properties of graphs arising from bridging and derive lower bounds for their least positive eigenvalue. As an illustration we present a census of graphs with a unique 1-factor on $m\le 6$ vertices and determine their positive and negative invertibility.