arXiv 2108.005922021-08-02math.CO

Smith Normal Form与图的广义谱刻划

Smith Normal Form and the Generalized Spectral Characterization of Graphs

Lihong Qiu, Wei Wang, Wei Wang, Hao Zhang

备注：: 14 pages

摘要：

图的谱刻画是谱图论中的一个重要课题，近年来受到了研究者的广泛关注。一般来说，很难用一个给定的图来表示它的谱。最近，Wang [10]给出了图由广义谱决定的一个简单的算术条件.设$G$是一个图，其邻接矩阵$A$在$n$的顶点上，$W=[e,Ae,\ldots,A^{n-1}e]$（$e$是全一向量）是$G$的行走矩阵. Wang [10]的一个定理指出，如果$2^{-\lfloor n/2\rfloor}\det W$（它总是一个整数）是奇数且无平方，则$G$由广义谱确定。在本文中，我们找到了一个新的和短的路线，导致上述定理的一个更强的版本。利用$G$的行走矩阵的Smith标准形，得到了该结果。所提出的方法提供了一个新的见解，在处理问题的广义谱特征的图。

英文摘要：

Spectral characterization of graphs is an important topic in spectral graph theory, which has received a lot of attention from researchers in recent years. It is generally very hard to show a given graph to be determined by its spectrum. Recently, Wang [10] gave a simple arithmetic condition for graphs being determined by their generalized spectra. Let $G$ be a graph with adjacency matrix $A$ on $n$ vertices, and $W=[e,Ae,\ldots,A^{n-1}e]$ ($e$ is the all-one vector) be the walk-matrix of $G$. A theorem of Wang [10] states that if $2^{-\lfloor n/2\rfloor}\det W$ (which is always an integer) is odd and square-free, then $G$ is determined by the generalized spectrum. In this paper, we find a new and short route which leads to a stronger version of the above theorem. The result is achieved by using the Smith Normal Form of the walk-matrix of $G$. The proposed method gives a new insight in dealing with the problem of generalized spectral characterization of graphs.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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