So关于p^aq阶整数循环图猜想的证明
So's Conjecture for Integral Circulant Graphs of Order p^aq
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中文总结 AI 辅助
本文证明了So猜想对p^a q阶整数循环图成立，通过谱计数测度处理重合特征值，并利用强归纳与连通分量分解恢复完整除数集。
中文摘要 AI 辅助
So猜想：对于固定的正整数n，n阶整数循环图的普通邻接谱决定其除数集。我们证明了当图的阶数为p^a乘以q（其中p和q为素数，p小于q，a至少为1）时该猜想成立。为了处理由不同最大公约数类产生的重合特征值，我们使用了一种谱计数测度。对于连通图，一个精确恒等式恢复了除数集中由1和q组成的部分（若存在），连同p^(a-1)乘以q阶图的计数测度。强归纳法及分解为连通分量随后恢复了完整的除数集，包括p等于2和离散图的情形。
英文摘要
So conjectured that, for a fixed positive integer n, the ordinary adjacency spectrum of an integral circulant graph of order n determines its divisor set. We prove this for graphs of order p to the power a times q, where p and q are primes with p less than q and a is at least one. To handle coincident eigenvalues arising from distinct greatest-common-divisor classes, we use a spectral counting measure. For connected graphs, an exact identity recovers the part of the divisor set consisting of one and q, when present, together with the counting measure for a graph of order p to the power a minus one times q. Strong induction and decomposition into connected components then recover the full divisor set, including the case p equals two and disconnected graphs.
发表机构
- School of Mathematics and Statistics, Weifang University（潍坊学院数学与统计学院）
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