Some short notes on oriented line graphs and related matrices

TL;DR

This paper explores the spectral properties of oriented line graphs and related matrices, providing explicit characteristic polynomials and applications to graph coloring, advancing understanding of graph spectra and their combinatorial implications.

Contribution

It determines the characteristic polynomial of the $z$-Hermitian adjacency matrix for regular graphs, generalizing known cases and linking spectral properties to graph coloring.

Findings

Derived the characteristic polynomial for the $z$-Hermitian adjacency matrix of oriented line graphs.

Connected spectral properties to the Ihara zeta function and graph coloring.

Provided explicit formulas for special cases including Hermitian adjacency matrices.

Abstract

Oriented line graph, introduced by Kotani and Sunada (2000), is closely related to Hashimato's non-backtracking matrix (1989). It is known that for regular graphs $G$, the eigenvalues of the adjacency matrix of the oriented line graph $\vec{L}(G)$ of $G$ are the reciprocals of the poles of the Ihara zeta function of $G$. We determine the characteristic polynomial of the $z$-Hermitian adjacency matrix of $\vec{L}(G)$ for each $z\in \mathbb{C}$ and $d$-regular graph $G$ with $d\geq 3$. Special cases of this matrix include the Hermitian adjacency matrix of $\vec{L}(G)$ and the adjacency matrix of the underlying undirected graph of $\vec{L}(G)$. We also exhibit an application to star coloring of graphs.