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.
From: Bilal Ahmad Rather [view email]
[v1]
Sat, 21 Mar 2026 21:02:48 UTC (24 KB)
[v2]
Wed, 16 Sep 2026 09:08:44 UTC (25 KB)
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