Seidel Laplacian and Seidel Signless Laplacian Spectrum of the Zero-divisor Graph on the Ring of Integers Modulo &lt;img src=image/13424080_01.gif&gt;

Sr.Magie Jose,Magi P M,Anjaly Kishore

doi:10.13189/ms.2021.090607

Abstract

Let <img src=image/13424080_02.gif> be a simple graph of order <img src=image/13424080_03.gif> and let <img src=image/13424080_04.gif> be the Seidel matrix of <img src=image/13424080_02.gif>, defined as <img src=image/13424080_05.gif> where <img src=image/13424080_06.gif> if the vertices <img src=image/13424080_07.gif> and <img src=image/13424080_08.gif> are adjacent and <img src=image/13424080_09.gif> if the vertices <img src=image/13424080_07.gif> and <img src=image/13424080_08.gif> are not adjacent and <img src=image/13424080_10.gif> if <img src=image/13424080_11.gif>. Let <img src=image/13424080_12.gif> be the diagonal matrix where <img src=image/13424080_13.gif> denotes the degree of the <img src=image/13424080_14.gif> vertex of <img src=image/13424080_02.gif>. The Seidel Laplacian matrix of a graph <img src=image/13424080_02.gif> is defined as <img src=image/13424080_15.gif> and the Seidel signless Laplacian matrix of a graph <img src=image/13424080_02.gif> is defined as <img src=image/13424080_16.gif>. The zero-divisor graph of a commutative ring <img src=image/13424080_17.gif>, denoted by <img src=image/13424080_18.gif>, is a simple undirected graph with all non-zero zero-divisors as vertices and two distinct vertices <img src=image/13424080_19.gif> are adjacent if and only if <img src=image/13424080_20.gif>. In this paper, we find the Seidel polynomial and Seidel Laplacian polynomial of the join of two regular graphs using the concept of schur complement and coronal of a square matrix. Also we describe the computation of the Seidel Laplacian and Seidel signless Laplacian eigenvalues of the join of more than two regular graphs, using the well known Fiedler's lemma and apply these results to describe these eigenvalues for the zero-divisor graph on <img src=image/13424080_21.gif>. Further we find the Seidel Laplacian and Seidel signless Laplacian spectrum of the zero-divisor graph of <img src=image/13424080_21.gif> for some values of <img src=image/13424080_03.gif>, say <img src=image/13424080_22.gif>, where <img src=image/13424080_23.gif> are distinct primes. We also prove that 0 is a simple Seidel Laplacian eigenvalue of <img src=image/13424080_24.gif>, for any <img src=image/13424080_03.gif>.

Highlights

Throughout this paper, G denotes a simple, finite, undirected and connected graph
It is very interesting and challenging that the combinatorial and spectral properties of zero-divisor graphs can be studied in terms of its compressed graph
The commutative ring Zp is an integral domain for any prime p and so it is quite trivial to study its zero-divisor graph

Summary

Introduction

Throughout this paper, G denotes a simple, finite, undirected and connected graph. If G has n vertices, the adjacency p q r s ¡ matrix, A G aij n¢n where, aij 1 if vi 6vj and aij 0 otherwise. G di is is denoted by specSpGq. 918 Seidel Laplacian and Seidel Signless Laplacian Spectrum of the Zero-divisor Graph on the Ring of Integers Modulo n and the Seidel signless Laplacian matrix of a graph G is defined as SL pGq DSpGq SpGq. For a complete graph Kn, and For a null graph, Kn, and SLpKnq Jn ¡ nIn SL pKnq p2 ¡ nqIn ¡ Jn. SLpKnq nIn ¡ Jn SL pKnq pn ¡ 2qIn Jn. Definition 2.1. The generalized join of H1, H2, ..., Hn denoted by G H1, H2, ..., Hn , is obtained by replacing each vertex i of G by the graph Hi and inserting all or none of the possible edges between Hi and Hj if i and j are adjacent in G or not. It is very interesting and challenging that the combinatorial and spectral properties of zero-divisor graphs can be studied in terms of its compressed graph

Fiedler’s Lemma and its generalisation

Generalization of Fiedler’s Lemma

The zero-divisor graph Γ pZnq

Seidel signless Laplacian spectrum of the join of regular graphs

Conclusion