Block Structure and Spectrum of Zero-Divisor Graphs of Lipschitz Quaternion Rings Modulo $n$

Abstract

We study the adjacency spectra of zero-divisor graphs associated with Lipschitz quaternion rings modulo $n$. For an odd prime $p$, the identification ${\mathbb{L}}_p\cong M_2({\mathbb{F}}_p)$ gives a kernel--image partition of the nonzero singular matrices and an explicit block representation of the adjacency matrix. This structure leads to a spectral decomposition, closed expressions for the spectral radius, and information on rank, nullity, eigenvalue multiplicities, and graph energy. For powers of $2$, we determine the graph at $n=2$ and use square-zero ideals in ${\mathbb{L}}_{2^t}$ to construct large complete subgraphs and obtain corresponding spectral and energy bounds. The resulting framework provides a unified structural approach to the odd-prime and two-adic cases and reduces large adjacency-matrix computations to substantially smaller algebraic models.