On Two Laplacian Matrices for Skew Gain Graphs

TL;DR

This paper introduces two types of Laplacian matrices for skew gain graphs, explores their properties, and proves a matrix tree theorem for the $g$-Laplacian in the context of fields of characteristic zero.

Contribution

It defines and analyzes Laplacian and $g$-Laplacian matrices for skew gain graphs, establishing a matrix tree theorem for the $g$-Laplacian.

Findings

Defined Laplacian and $g$-Laplacian matrices for skew gain graphs

Proved the matrix tree theorem for the $g$-Laplacian matrix

Extended classical graph theory results to skew gain graph context

Abstract

Let $G=(V,\overrightarrow{E})$ be a graph with some prescribed orientation for the edges and $\Gamma$ be an arbitrary group. If $f\in \mathrm{Inv}(\Gamma)$ be an anti-involution then the skew gain graph $\Phi_f=(G,\Gamma,\varphi,f)$ is such that the skew gain function $\varphi:\overrightarrow{E}\rightarrow \Gamma$ satisfies $\varphi(\overrightarrow{vu})=f(\varphi(\overrightarrow{uv}))$. In this paper, we study two different types, Laplacian and $g$-Laplacian matrices for a skew gain graph where the skew gains are taken from the multiplicative group $F^\times$ of a field $F$ of characteristic zero. Defining incidence matrix, we also prove the matrix tree theorem for skew gain graphs in the case of the $g$-Laplacian matrix.