### Journals Information

**
Mathematics and Statistics Vol. 9(6), pp. 917 - 926 DOI: 10.13189/ms.2021.090607 Reprint (PDF) (287Kb) **

## Seidel Laplacian and Seidel Signless Laplacian Spectrum of the Zero-divisor Graph on the Ring of Integers Modulo

**Magi P M ^{1}^{,*}, Sr.Magie Jose ^{1}, Anjaly Kishore ^{2}**

^{1}Department of Mathematics, St.Mary’s College, Thrissur, Kerala, India

^{2}Department of Mathematics, Vimala College, Thrissur, Kerala, India

**ABSTRACT**

Let be a simple graph of order and let be the Seidel matrix of , defined as where if the vertices and are adjacent and if the vertices and are not adjacent and if . Let be the diagonal matrix where denotes the degree of the vertex of . The Seidel Laplacian matrix of a graph is defined as and the Seidel signless Laplacian matrix of a graph is defined as . The zero-divisor graph of a commutative ring , denoted by , is a simple undirected graph with all non-zero zero-divisors as vertices and two distinct vertices are adjacent if and only if . 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 . Further we find the Seidel Laplacian and Seidel signless Laplacian spectrum of the zero-divisor graph of for some values of , say , where are distinct primes. We also prove that 0 is a simple Seidel Laplacian eigenvalue of , for any .

**KEYWORDS**

Seidel Laplacian Matrix, Zero-divisor Graph

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Magi P M , Sr.Magie Jose , Anjaly Kishore , "Seidel Laplacian and Seidel Signless Laplacian Spectrum of the Zero-divisor Graph on the Ring of Integers Modulo ," Mathematics and Statistics, Vol. 9, No. 6, pp. 917 - 926, 2021. DOI: 10.13189/ms.2021.090607.

(b). APA Format:

Magi P M , Sr.Magie Jose , Anjaly Kishore (2021). Seidel Laplacian and Seidel Signless Laplacian Spectrum of the Zero-divisor Graph on the Ring of Integers Modulo . Mathematics and Statistics, 9(6), 917 - 926. DOI: 10.13189/ms.2021.090607.