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arXiv 1902.00747math.CO
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关于某些符号图的特征值

On the eigenvalues of some signed graphs

M. Souri, F. Heydari, M. Maghasedi

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摘要：

设$G$是一个简单图，$A(G)$是$G$的邻接矩阵.矩阵$S(G) = J -I -2A(G)$被称为$G$的赛德尔矩阵，其中$I$是单位矩阵，而$J$是所有元素都等于1的方阵。显然，如果$G$是一个没有孤立点的$n$阶图，则$G$的Seidel矩阵也是一个符号完全图$K_n$的邻接矩阵，其负边诱导$G$。本文研究了完全多部图$K_{n_1,\ldots,n_k}$的Seidel特征值，并研究了它的Seidel特征多项式.我们表明，如果有至少三个部分的大小$n_i$，对于一些$i=1,\ldots,k$，那么$K_{n_1,\ldots,n_k}$是确定的，直到切换，其赛德尔谱。

英文摘要：

Let $G$ be a simple graph and $A(G)$ be the adjacency matrix of $G$. The matrix $S(G) = J -I -2A(G)$ is called the Seidel matrix of $G$, where $I$ is an identity matrix and $J$ is a square matrix all of whose entries are equal to 1. Clearly, if $G$ is a graph of order $n$ with no isolated vertex, then the Seidel matrix of $G$ is also the adjacency matrix of a signed complete graph $K_n$ whose negative edges induce $G$. In this paper, we study the Seidel eigenvalues of the complete multipartite graph $K_{n_1,\ldots,n_k}$ and investigate its Seidel characteristic polynomial. We show that if there are at least three parts of size $n_i$, for some $i=1,\ldots,k$, then $K_{n_1,\ldots,n_k}$ is determined, up to switching, by its Seidel spectrum.

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。

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