On the Characteristic Polynomial of Skew Gain Graphs

TL;DR

This paper investigates the spectral properties of skew gain graphs, a generalization of gain graphs involving anti-involutions, and explores their adjacency matrices and characteristic polynomials.

Contribution

It introduces skew gain graphs with involutive automorphisms, analyzes their adjacency matrices, and discusses their spectra, extending the theory of gain graphs.

Findings

Derived characteristic polynomials for skew gain graphs

Analyzed spectra of specific skew gain graphs

Connected weighted graphs as special cases

Abstract

Gain graphs are graphs where the edges are given some orientation and labeled with the elements (called gains) from a group so that gains are inverted when we reverse the direction of the edges. Generalizing the notion of gain graphs, skew gain graphs have the property that the gain of a reversed edge is the image of edge gain under an anti-involution. In this paper, we deal with the adjacency matrix of skew gain graphs with involutive automorphism on a field of characteristic zero and their charactersitic polynomials. Spectra of some particular skew gain graphs are also discussed. Meanwhile it is interesting to note that weighted graphs are particular cases of skew gain graphs.