Universal adjacency spectrum of zero divisor graph on the ring and its complement

Abstract

For a commutative ring R with unity, the zero divisor graph is an undirected graph with all non-zero zero divisors of R as vertices and two distinct vertices u and v are adjacent if and only if uv = 0. For a simple graph G with the adjacency matrix A and degree diagonal matrix D, the universal adjacency matrix is where I is identity matrix and J is all-ones matrix. For a graph H on k vertices and a family of vertex disjoint regular graphs we determine eigenpairs of the universal adjacency matrix of H-join of in terms of eigenpairs of the adjacency matrix of Hi, and a symmetric matrix of order k. For a non-prime integer n > 3, we obtain eigenpairs of and As an application, we also discuss the adjacency, Seidel, Laplacian and signless Laplacian spectra of both and Lastly, we determine the characteristic polynomial of for prime p and integer m > 1 (except for with ).

Keywords

• zero divisor graph
• universal adjacency matrix
• eigenvector
• eigenvalue
• complement