Spectral properties of the Cayley Graphs of split metacyclic groups

2016/09/20 by Rajeevsarathy, Kashyap, Sarkar, Siddhartha, Lakshmivarahan, S. +1
#05C50 (Secondary) #68R10 (Primary) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1609.06022

Abstract

Let Γ(G,S) denote the Cayley graph of a group G with respect to a set S ⊂ G. In this paper, we analyze the spectral properties of the Cayley graphs T_m,n,k = Γ(ℤ_m \ltimes_k ℤ_n, \(± 1,0),(0,± 1)\), where m,n ≥ 3 and k^m ≡ 1 \pmodn. We show that the adjacency matrix of T_m,n,k, upto relabeling, is a block circulant matrix, and we also obtain an explicit description of these blocks. By extending a result due to Walker-Mieghem to Hermitian matrices, we show that T_m,n,k is not Ramanujan, when either m > 8, or n ≥ 400.

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