由其Seidel谱确定的直到切换的完全多部图
Complete multipartite graphs that are determined, up to switching, by their Seidel spectrum
摘要：
已知完全多部图是由其距离谱而不是由其邻接谱决定的。一个图$G$在多个顶点上的赛德尔谱并不决定该图，因为通过赛德尔转换从$G$得到的任何图都具有相同的赛德尔谱。我们认为$G$是由它的赛德尔谱，直到开关，如果任何具有相同的频谱的图切换等价于一个图同构于$G$。证明了具有与完全$k$ -部图相同的谱的图是完全$k$-部图的交换等价图，并且如果不同的划分集的大小是$p_1,\ldots, p_l$，并且每个大小的划分集至少有三个$p_i$，$i=1,\ldots, l$，则$G$由其Seidel谱确定，直到交换.讨论了完全三部图由其Seidel谱确定的充分条件，并对顶点数大于18的完全三部图提出了一个猜想。
英文摘要：
It is known that complete multipartite graphs are determined by their distance spectrum but not by their adjacency spectrum. The Seidel spectrum of a graph $G$ on more than one vertex does not determine the graph, since any graph obtained from $G$ by Seidel switching has the same Seidel spectrum. We consider $G$ to be determined by its Seidel spectrum, up to switching, if any graph with the same spectrum is switching equivalent to a graph isomorphic to $G$. It is shown that any graph which has the same spectrum as a complete $k$-partite graph is switching equivalent to a complete $k$-partite graph, and if the different partition sets sizes are $p_1,\ldots, p_l$, and there are at least three partition sets of each size $p_i$, $i=1,\ldots, l$, then $G$ is determined, up to switching, by its Seidel spectrum. Sufficient conditions for a complete tripartite graph to be determined by its Seidel spectrum are discussed, and a conjecture is made on complete tripartite graphs on more than 18 vertices.
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