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.

翻译：我们研究了由Lipschitz四元数环模\(n\)导出的零因子图的邻接矩阵。对于奇素数\(p\)，利用同构\(\LL_p\cong M_2(\F_p)\)，我们根据核-像类型对顶点进行分类，并证明邻接矩阵具有作为射影关联矩阵的膨胀的块结构。这可以在类常数子空间上生成一个简化矩阵，并给出零化度和特征值\(-1\)重数下界的精确公式，以及通过均衡划分得到的谱半径的闭合表达式。对于二进族，我们精确确定了\(n=2\)时的图，并证明当\(t\ge 2\)时，图\(G_{2^t}\)包含由理想滤波导出的大团，这些团给出了谱半径的明确下界。我们还探讨了对图能量的影响，并提供了邻接矩阵的系统性构造方法。