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

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.