[論文レビュー] Block Structure and Spectrum of Zero-Divisor Graphs of Lipschitz Quaternion Rings Modulo \(n\)

Bilal Ahmad Rather|arXiv (Cornell University)|Mar 21, 2026

Graph theory and applications被引用数 0

ひとこと要約

The paper analyzes the adjacency matrices of zero-divisor graphs from Lipschitz quaternion rings modulo n, deriving a block-structured matrix model for odd primes and exact spectral information, plus results for the 2-power case.

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.

研究の動機と目的

• Motivate the study of zero-divisor graphs arising from Lipschitz quaternion rings modulo n and their adjacency matrices.
• Classify and model the adjacency structure when n is an odd prime p, via kernel–image types linked to M2(Fp).
• Develop a block-structured (equitable) matrix representation to facilitate spectral analysis, including nullity and spectral radius results.
• Extend the analysis to the two-adic family, especially n = 2 and n = 2^t, identifying clique structures and providing bounds for spectrum and energy.
• Provide constructive methods for building the adjacency matrix and discuss implications for graph energy.

提案手法

• Identify Lipschitz quaternion ring L_n and its zero-divisor graph G_n with adjacency a-xy=0 or yx=0.
• For odd prime p, use L_p ≅ M2(F_p) to classify nonzero singular matrices by kernel-image pairs (L,M) from the projective line, forming a type partition with (p+1)^2 classes.
• Show A_p is permutation similar to H_p ⊗ J_{p-1} − D_p ⊗ I_{p-1}, yielding a block model for A_p.
• Prove that off-diagonal blocks are J_{p-1} or O_{p-1} according to incidence conditions, and diagonal blocks are J_{p-1}−I_{p-1} when L=M.
• Construct an invariant subspace U spanned by class-constant vectors and W_{L,M} subspaces; derive the reduced characteristic polynomial χ_{A_p}(λ) = χ_{B_p}(λ)·λ^{p(p+1)(p−2)}·(λ+1)^{(p+1)(p−2)}.

実験結果

リサーチクエスチョン

• RQ1 How can the adjacency structure of G_p (Lipschitz quaternions modulo p) be captured by a block decomposition using kernel-image types?
• RQ2 What are the exact spectral properties (nullity, multiplicities, spectral radius) of the adjacency matrix A_p for odd primes p?
• RQ3 How does the two-adic (n=2^t) case differ structurally and spectrally from the odd-prime case, and what lower bounds can be established for the spectrum and energy?
• RQ4 Can an equitable partition lead to a reduced matrix whose spectrum determines the main spectral features of A_p?
• RQ5 What is the full adjacency-construction algorithm that avoids enumerating all singular matrices, and how do the results illustrate the energy of Γ(L_n) for n=2^t?

主な発見

• For odd primes p, A_p is permutation similar to H_p ⊗ J_{p-1} − D_p ⊗ I_{p-1} with a block structure determined by kernel-image types.
• The type partition has (p+1)^2 blocks each of size p−1, yielding total vertex count p^3 + p^2 − p − 1.
• G_p is biregular: diagonal-type vertices (L=M) have degree 2p^2 − p − 2, non-diagonal-type vertices have degree 2p^2 − p − 1.
• The spectrum includes a forced component from class-constant subspace U, with χ_{A_p}(λ)=λ^{p(p+1)(p−2)}(λ+1)^{(p+1)(p−2)}χ_{B_p}(λ).
• A full description of the graph at n=2 is provided, and for t≥2, large cliques inside Γ(L_{2^t}) yield explicit lower bounds for the spectral radius and energy considerations.
• An algorithmic construction of A_p is given, along with numerical examples and illustrations.