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Some short notes on oriented line graphs and related matrices

Jacob Antony, Cyriac Antony, Jinitha Varughese, Bloomy Joseph

TL;DR

The paper addresses the problem of determining the z-Hermitian spectrum of the oriented line graph L(G) of a d-regular graph and its connection to Hashimoto's non-backtracking matrix and the Ihara zeta function. It uses a unitary similarity to a block-diagonal form to derive an explicit z-Hermitian characteristic polynomial for L(G) in terms of the base graph's eigenvalues lambda_i and the degree d, revealing a product structure that depends on z. It also provides a direct proof for the spectrum of the undirected version L^*(G) and discusses consequences for star coloring via Locally Bijective Homomorphisms, including divisibility and factorization results. Overall, the work deepens the spectral understanding of oriented line graphs, links to zeta-function poles, and yields practical implications for graph coloring problems.

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.

Some short notes on oriented line graphs and related matrices

TL;DR

The paper addresses the problem of determining the z-Hermitian spectrum of the oriented line graph L(G) of a d-regular graph and its connection to Hashimoto's non-backtracking matrix and the Ihara zeta function. It uses a unitary similarity to a block-diagonal form to derive an explicit z-Hermitian characteristic polynomial for L(G) in terms of the base graph's eigenvalues lambda_i and the degree d, revealing a product structure that depends on z. It also provides a direct proof for the spectrum of the undirected version L^*(G) and discusses consequences for star coloring via Locally Bijective Homomorphisms, including divisibility and factorization results. Overall, the work deepens the spectral understanding of oriented line graphs, links to zeta-function poles, and yields practical implications for graph coloring problems.

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 , the eigenvalues of the adjacency matrix of the oriented line graph of are the reciprocals of the poles of the Ihara zeta function of . We determine the characteristic polynomial of the -Hermitian adjacency matrix of for each and -regular graph with . Special cases of this matrix include the Hermitian adjacency matrix of and the adjacency matrix of the underlying undirected graph of . We also exhibit an application to star coloring of graphs.

Paper Structure

This paper contains 5 sections, 9 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $z$-Hermitian Spectrum of $\hbox{\boldmath$L$}(G)$
  3. Consequence to star coloring
  4. Disclosure of Interests.
  5. Direct proof for spectrum of $L^*(G)$

Key Result

theorem thmcountertheorem

Let $G$ be a connected $d$-regular graph with $n$ vertices, $m$ edges and eigenvalues $\lambda_1,\lambda_2,\dots,\lambda_n$, where $d\geq 3$. Then, the non-backtracking matrix of $G$ is unitarily similar to the following block-diagonal matrix $C$, where $-1$ and $1$ are repeated $m-n$ times, and $\{\theta_i,\theta_i'\}=\{(\lambda/2)+\sqrt{(\lambda/2)^2-(d-1)},(\lambda/2)-\sqrt{(\lambda/2)^2-(d-1

Figures (1)

  • Figure 1: An example of the oriented line graph operation.

Theorems & Definitions (11)

  • theorem thmcountertheorem: Lubetzky and Peres lubetzky_peres
  • theorem thmcountertheorem
  • proof : Proof overview
  • corollary thmcountercorollary
  • corollary thmcountercorollary
  • corollary thmcountercorollary
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • ...and 1 more