A study on the spectrum of zero divisor graph on the ring of integers modulo n

Abstract

Graph Theory is an important branch of Discrete Mathematics, which is a key tool to model network systems involved in major domains of real life. Graph Theory extends its countless applications to various walks of science like Net- work Theory, Operational Research, Chemistry, Quantum Physics, Biology, Economics, Artificial Intelligence, Sociology and so on. Exploring algebraic struc- tures through graph theory has become a captivating research field over the past three decades. Researchers have extensively studied graphs associated with algebraic structures such as groups and rings, viz Cayley graphs, power graphs, zero-divisor graphs and co-maximal graphs, etc. Such study provides intercon- nections between Algebra and Graph Theory. The zero divisor graph Γ p R q of a commutative ring R is the simple undirected graph with vertices non-zero zero- divisors of R and two distinct vertices x, y are adjacent if xy 0. This thesis focuses on the study of different matrices associated with the zero divisor graph on the ring of integers modulo n and explores its spectra.Usually, the eigenvalues of a graph can be computed by finding the roots of its characteristic polynomial. But there is no algebraic method to solve a polynomial equation of degree greater than or equal to five. This makes the computation of spectrum of graphs tedious. However, for a graph with large size and complicated combinatorial structure, the determination of spectra is really challenging. Sometimes, it becomes a convenient practice that the spectrum of a fairly large graph can be described in terms of the spectra of smaller graphs using some simple graph operations, like union, join, corona, edge corona etc.The analysis of the adjacency matrix of the zero divisor graph on Z n , for n p 2 q 2 , p 2 q, p k , k ¡ 1, where p, q are distinct primes, leads to some intriguing results about the graph parameters of these graphs as well as their characteristic polynomials.Analogous to the Laplacian and signless Laplacian matrix of a graph, the definition of distance Laplacian and distance signless Laplacian matrix was introduced and studied by M. Aouchiche and P. Hansen. In this thesis, the study on the distance, distance Laplacian and distance signless Laplacian spectrum of Γ p Z n q has been initiated. The eigenvalues of the distance and distance Laplacian matrix of Γ p Z n q for some values of n, are found along with multiplicities, by direct computation using matrix tools. The distance Laplacian eigenvalues ofΓ p Z p k q , where p is any prime and k ¡ 1 is any positive integer, are completely explored with multiplicities. Also, a general method is proposed for finding the characteristic polynomial of the distance and distance Laplacian matrix of Γ p Z n q for any n.H.S. Ramane et al defined Seidel Laplacian and Seidel signless Laplacian matrix of graphs. In this thesis, Seidel, Seidel Laplacian and Seidel signless Laplacian spectrum of the generalised union of regular graphs is investigated and extended these results to the zero divisor graph on the ring of integers modulo n.