The Laplacian and signless Laplacian spectrum of semi-Cayley graphs over abelian groups

Abstract

In this paper, a formula of the Laplacian and signless Laplacian spectrum of semi-Cayley graphs over abelian groups is given. As applications of our main result, special formulae of Laplacian and signless Laplacian spectrum are also given for two classes of semi-Cayley graphs (one matching bi-Cayley graphs and the join of two Cayley graphs over isomorphic abelian groups). In particular, a method to construct Laplacian and signless Laplacian integral semi-Cayley graphs is obtained.

On the Laplacian and signless Laplacian polynomials of graphs with semiregular automorphisms

“…This result was improved for n-Cayley graphs over arbitrary groups in 2013 by the present author and Taeri [1]. In 2015, the Laplacian and signless Laplacian spectrum of semi-Cayley graphs over abelian groups is computed by Gao et al [12]. It seems that their argument cannot be extended to non-abelian groups or n-Cayley graphs for n ≥ 3.…”

“…Since every semi-Cayley graph over an abelian group is a quasiabelian semi-Cayley graph and all character degrees of abelian groups are 1, one can get the main result of [12], see [12,Theorem 1], by putting α = 1, −1 in the following corollary.…”

“…n-Cayley graphs, in particular when n = 2 or n = 3, have been playing an important role in many classical fields of graph theory, such as strongly regular graphs [19,20,22,23,27], symmetry properties of graphs [7,8,21] and the spectrum of graphs [1,3,4,11,12].…”

Eigenvalues of Cayley graphs

“…In [122], Gao et al obtained formulas for the Laplacian and signless Laplacian spectra of bi-Cayley graphs on abelian groups. As applications of their main result, special formulas for the Laplacian and signless Laplacian spectra are also given for two classes of bi-Cayley graphs, namely one-matching bi-Cayley graphs and the join of two Cayley graphs over isomorphic abelian groups.…”

On the Eigenvalues of N-Cayley Graphs: A Survey

“…n-Cayley graphs, in particular when n = 2 or n = 3, have played an important role in many classical fields of graph theory, such as strongly regular graphs [19,22,23,24,25], automorphisms [2,15,28], isomorphisms [3,5], symmetry properties of graphs [10,11,20] and the spectrum of graphs [1,4,8,12,13]. In this paper, we review recent results and future directions of some problems related to the spectrum of n-Cayley graphs.…”