On Laplacian spectrum of unitary Cayley graphs

S Pirzada,M Afkhami,Z Barati

doi:10.2478/ausi-2021-0011

Abstract

Abstract Let R be a commutative ring with unity 1 ≠ 0 and let R× be the set of all unit elements of R. The unitary Cayley graph of R, denoted by GR = Cay(R, R×), is a simple graph whose vertex set is R and there is an edge between two distinct vertices x and y of R if and only if x − y ∈ R×. In this paper, we determine the Laplacian and signless Laplacian eigenvalues for the unitary Cayley graph of a commutative ring. Also, we compute the Laplacian and signless Laplacian energy of the graph GR and its line graph.

Highlights

We consider finite commutative rings R with unit element 1 = 0
The Laplacian eigenvalues of the direct product of two regular graphs are listed in the following theorem
(c) From case (3.3), |R×1 × R×2 × · · · × R×t−1||Rt| + λA|R×t | − λA|Rt| with multiplicity ( i∈A)(ft − 1) is a Laplacian eigenvalue of GR, for all A {1, 2, . . . , t − 1}

Summary

Introduction

We consider finite commutative rings R with unit element 1 = 0. Let R× be the set of all unit elements of R. We know that an Artinian ring R can be written as R =∼ R1 × · · · × Rt, where Ri is a finite local ring with maximal ideal Mi, for all 1 i t. This decomposition is unique up to permutation of factors. We. Computing Classification System 1998: G.2.2 Mathematics Subject Classification 2010: 05C22, 05C50, 05C76 Key words and phrases: unitary Cayley graph, Laplacian spectrum, signless Laplacian spectrum, Laplacian energy

Ki and fi

Since the Laplacian spectrum of GR is