2015/03/05 by Aliréza Abdollahi, Alireza Abdollahi, Abdollahi, Alireza +4
Engineering · Mathematics · #05C25 #05C50 #15A18 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Group Theory (math.GR) #Spectral Theory (math.SP) #graph theory and CDMA systems #math.CO #math.GR #math.SP #msc:05C25 #msc:05C50 #msc:15A18
paper · pdf · doi:10.48550/arxiv.1503.01541
The proof of Proposition 2.1 is corrected. The proof of Theorem 2.4 is corrected
openalex publication_date 2015/03/05 · arxiv created 2015/05/02 · arxiv updated 2015/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a finite group, and S be a subset of G∖\1\ such that S=S-1. Suppose that Cay(G,S) is the Cayley graph on G with respect to the set S which is the graph whose vertex set is G and two vertices a,b∈ G are adjacent if and only if ab-1∈ S. The adjacency spectrum Spec(Γ) of a graph Γ is the multiset of eigenvalues of its adjacency matrix. A graph Γ is called "determined by its spectrum" (or for short DS) whenever if a graph Γ' has the same spectrum as Γ, then Γ≅ Γ'. We say that the group G is DS (Cay-DS, respectively) whenever if Γ is a Cayley graph over G and Spec(Γ)=Spec(Γ') for some graph (Cayley graph, respectively) Γ', then Γ≅ Γ'. In this paper, we study finite DS groups and finite Cay-DS groups. In particular we prove that all finite DS groups are solvable and all Sylow p-subgroups of a finite DS group is cyclic for all p≥ 5. 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.