Block Structure and Spectrum of Zero-Divisor Graphs of Lipschitz Quaternion Rings Modulo \(n\)
This work addresses a specific algebraic graph theory problem, focusing on properties of zero-divisor graphs in non-commutative rings, which is incremental within its niche domain.
The paper tackles the problem of analyzing the adjacency matrices and spectra of zero-divisor graphs from Lipschitz quaternion rings modulo n, resulting in a block structure for odd primes with formulas for nullity bounds and eigenvalue multiplicities, and for powers of two, it identifies cliques that give lower bounds on the spectral radius.
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.