Groups all of whose undirected Cayley graphs are determined by their spectra

Abstract

The adjacency spectrum [Formula: see text] of a graph [Formula: see text] is the multiset of eigenvalues of its adjacency matrix. Two graphs with the same spectrum are called cospectral. A graph [Formula: see text] 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 [Formula: see text] is Cay-DS if every two cospectral Cayley graphs of [Formula: see text] 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 [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. We also give several infinite families of non-Cay-DS solvable groups. In particular we prove that there exist two cospectral non-isomorphic [Formula: see text]-regular Cayley graphs on the dihedral group of order [Formula: see text] for any prime [Formula: see text].