Block Structure and Spectrum of Zero-Divisor Graphs of Lipschitz Quaternion Rings Modulo \(n\)

TL;DR

This paper analyzes the structure and spectrum of zero-divisor graphs from Lipschitz quaternion rings modulo n, revealing block structures, eigenvalue bounds, and spectral properties for various cases.

Contribution

It provides a detailed spectral analysis and block structure characterization of zero-divisor graphs of Lipschitz quaternion rings modulo n, including explicit formulas and bounds.

Findings

Adjacency matrix has a block structure as a blow-up of a projective incidence matrix.

Derived bounds for nullity and eigenvalue multiplicities, especially for eigenvalue -1.

Identified clique structures and spectral radius bounds for 2-adic cases.

Abstract

We investigate the adjacency matrices of zero-divisor graphs derived from Lipschitz quaternion rings modulo \(n\). For odd primes \(p\), utilizing the isomorphism \(\LL_p\cong M_2(\F_p)\), we categorize vertices by kernel-image type and demonstrate that the adjacency matrix possesses a block structure as a blow-up of a projective incidence matrix. This produces a reduced matrix on the class-constant subspace, with precise formula for the lower bound for the nullity and the multiplicity of the eigenvalue \(-1\), as well as a closed expression for the spectral radius through an equitable partition. For the two-adic family, we precisely ascertain the graph at \(n=2\) and demonstrate that for \(t\ge 2\), the graph \(G_{2^t}\) encompasses substantial cliques derived from the ideal filtering, which yield definitive lower bounds for the spectral radius. We also examine the implications for graph energy and provide a systematic construction of the adjacency matrix.