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

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.