笛卡尔积的穿孔邻接-度代数
Punctured adjacency-degree algebras of Cartesian products
- Shenzhen MSU–BIT University（深圳北理莫斯科大学）
- Guangdong Laboratory of Machine Perception and Intelligent Computing（广东省机器感知与智能计算实验室）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究笛卡尔积图的穿孔邻接-度代数，通过局部谱生成函数和仿射秩确定循环模维数，并给出汉明图等特例的闭式结果。
AI中文摘要：
对于连通正则图 G 和顶点 a，我们研究由 G-a 的邻接矩阵和度矩阵生成的代数及其由全一向量生成的循环模 P_a。我们的主定理确定了在所选根处具有等距划分的因子的笛卡尔积的 dim P_a。局部谱生成函数的归一化对数导数将因子划分为边界类。我们精确地识别了边界返回空间，并将 dim P_a 表示为加性谱纤维上的仿射秩之和。对于具有不同谱 Θ 的距离正则因子，这给出 dim P_a(F^{□m}) = |mΘ| - 1。对于两个不同完全图的幂的乘积，我们以闭式形式评估纤维公式。我们还确定了所有汉明图的完整穿孔代数：与压缩 Terwilliger 代数的相等性恰好发生在超立方体的维数至多为四以及更大字母表的维数至多为二时。对于距离正则图，邻接矩单独确定交阵列，并具有显式的有限重构。最后，笛卡尔稳定化子公式将度量损失与轨道分裂分开；在 Doob 图上，它们的距离分级缺陷恢复了 Shrikhande 因子的数量。
英文摘要：
For a connected regular graph G and a vertex a, we study the algebra generated by the adjacency and degree matrices of G-a and its cyclic module P_a generated by the all-ones vector. Our main theorem determines dim P_a for Cartesian products whose factors have equitable distance partitions at the chosen roots. A normalized logarithmic derivative of the local spectral generating function partitions the factors into boundary classes. We identify the boundary-return space exactly and express dim P_a as a sum of affine ranks on additive spectral fibres. For a distance-regular factor with distinct spectrum Θ, this gives dim P_a(F^{\square m}) = |mΘ| - 1. For products of powers of two distinct complete graphs, we evaluate the fibre formula in closed form. We also determine the full punctured algebras of all Hamming graphs: equality with the compressed Terwilliger algebra holds precisely in dimensions at most four for the hypercube and at most two for larger alphabets. For distance-regular graphs, adjacency moments alone determine the intersection array, with an explicit finite reconstruction. Finally, Cartesian stabilizer formulas separate metric loss from orbit splitting; on Doob graphs their distance-graded defect recovers the number of Shrikhande factors.
