# 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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