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.

Figures (4)

• Figure 1: A block representation of $G_3$.
• Figure 2: Block diagram of the equitable partition $V(G_p)=\mathcal{D}\sqcup\mathcal{O}$. The labels are the row sums of the quotient matrix $Q_p$. This diagram explains why the spectral radius can be computed from a $2\times 2$ matrix.
• Figure 3: The graph $G_2$.
• Figure 4: Mechanism of the two-adic energy bound.

Theorems & Definitions (29)

• Proposition 2.1
• Theorem 2.2: grau2017
• Lemma 2.3
• Definition 3.1
• Lemma 3.2
• Theorem 3.3
• Corollary 3.4
• Corollary 3.5
• Example 3.6
• Theorem 4.1