On Two Laplacian Matrices for Skew Gain Graphs

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 ${\Phi }_f=(G,\Gamma ,\phi ,f)$ is such that the skew gain function $\phi :\overrightarrow{E}\to \Gamma$ satisfies $\phi (\overrightarrow{vu})=f(\phi (\overrightarrow{uv}))$. 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.

15 pages