Hermitian adjacency matrices with at most three distinct eigenvalues

TL;DR

This work completely characterizes connected oriented graphs whose Hermitian adjacency matrix $H_\omega$ (with a primitive sixth root of unity) has exactly two distinct eigenvalues, yielding only four examples. It shows there are infinitely many regular tournaments with exactly three $H_\omega$-eigenvalues, constructed via skew-Hadamard matrices, and proves a two-eigenvalue classification extends to certain mixed graphs. The paper further investigates $H_\sigma$ for other roots of unity, revealing rich connections to signed graphs and design theory, and establishes limits for large $k$ while outlining open questions for intermediate cases.

Abstract

We study oriented graphs whose Hermitian adjacency matrices of the second kind have few eigenvalues. We give a complete characterization of the oriented graphs with two distinct eigenvalues, showing that there are only four such graphs. We extend this result to mixed graphs. We show that there are infinitely many regular tournaments with three distinct eigenvalues. We extend our main results to Hermitian adjacency matrices defined over other roots of unity.

Figures (1)

• Figure 1: The special oriented graphs from Theorem \ref{['Thm:2ev_Char']} (c) and (d).

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2: Eigenvalue interlacing
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof