Journal of the Ramanujan Mathematical Society

Volume 35, Issue 2, June 2020 pp. 159–175.

Spectral properties of Cayley graphs of split metacyclic groups

Authors: Kashyap Rajeevsarathy, Siddhartha Sarkar, Sivaramakrishnan Lakshmivarahan and Pawan Kumar Aurora
Author institution:Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462 066, Madhya Pradesh, India

Summary: 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 (ℤm ⋉ k ℤn, {(±1, 0), (0,±1)}), where m, n ≥ 3 and km ≡ 1 (mod n). We show that the adjacency matrix of T m,n,k , upto relabeling, is a block-circulant matrix, and obtain an explicit description of these blocks. By applying some known bounds on the eigenvalues of Hermitian matrices, we show that T m,n,k is not Ramanujan, when either m > 8, or n ≥ 400. Finally, we list the collection of all pairs in {(m, n) : 3 ≤ m ≤ 8 and 3 ≤ n < 400} for which there exist at least one k > 1 such that T m,n,k is Ramanujan.

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