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首页> 外文期刊>Journal of algebra and its applications >Groups all of whose undirected Cayley graphs are determined by their spectra
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Groups all of whose undirected Cayley graphs are determined by their spectra

机译：将所有其无向Cayley图由其光谱确定的组

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The adjacency spectrum Spec(G) of a graph G is the multiset of eigenvalues of its adjacency matrix. Two graphs with the same spectrum are called cospectral. A graph G is "determined by its spectrum" (DS for short) if every graph cospectral to it is in fact isomorphic to it. A group is DS if all of its Cayley graphs are DS. A group G is Cay-DS if every two cospectral Cayley graphs of G are isomorphic. In this paper, we study finite DS groups and finite Cay-DS groups. In particular we prove that a finite DS group is solvable, and every non-cyclic Sylow subgroup of a finite DS group is of order 4, 8, 16 or 9. We also give several infinite families of non-Cay-DS solvable groups. In particular we prove that there exist two cospectral non-isomorphic 6-regular Cayley graphs on the dihedral group of order 2p for any prime p >= 13.
机译：图G的邻接光谱Spec（G）是其邻接矩阵特征值的多集。具有相同光谱的两个图称为共光谱。如果与其共谱的每个图实际上都是同构的，则图G“由其频谱确定”（简称DS）。如果其所有Cayley图均为DS，则该组为DS。如果G的每两个同谱Cayley图是同构的，则G组为Cay-DS。在本文中，我们研究有限DS群和有限Cay-DS群。特别地，我们证明了有限DS组是可解的，并且有限DS组的每个非循环Sylow子组的阶数为4、8、16或9。我们还提供了数个无限的非Cay-DS可解基团。特别地，我们证明对于任何质数p> = 13，在2p阶二面体组上都存在两个共谱非同构6正则Cayley图。

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