On main eigenvalues of zero-divisor graphs of reduced rings

TL;DR

This paper explores the spectral properties of zero-divisor graphs of reduced rings, demonstrating they form infinite families with a fixed number of main eigenvalues, advancing understanding in spectral graph theory.

Contribution

It establishes that zero-divisor graphs of reduced rings and certain bipartite subgraphs have a specified number of main eigenvalues, providing new infinite families in spectral graph theory.

Findings

Zero-divisor graphs of reduced rings have exactly s main eigenvalues for any positive integer s.

Certain induced bipartite subgraphs also have exactly s main eigenvalues.

These graphs form infinite families with prescribed spectral properties.

Abstract

The problem of characterizing graphs with a prescribed number of main eigenvalues is a long-standing problem in spectral graph theory. Although some constructions are known, only a few produce infinite families of simple connected graphs with exactly $s \ge 2$ main eigenvalues. Zero-divisor graphs form a well-structured class of algebraic graphs whose spectra can be described explicitly using equitable partitions, making them a convenient setting to study main eigenvalues. In this paper, we prove that the zero-divisor graphs of reduced rings provide an infinite family of simple connected graphs with exactly $s$ main eigenvalues, and that certain induced bipartite subgraphs also have exactly $s$ main eigenvalues for any positive integer $s$.