On The Study Of Some Invariants Of Unit Graphs Of Rings

It is one of active topics in ring theory studying the properties and structure ofrings by using the properties of special elements in a ring.In recent decades,associating with a graph to a ring and using the resulting graph to study the properties and structure of rings is a hot topic.The unit graph of a ring R is the simple graph,denoted by G(R),whose vertex set is R,and in which two distinct vertices x and y are adjacent if and only if x(10)y is a unit of R.In this paper,we mainly study the unit graph of a ringThe first chapter introduces the research background of this thesis and related to some basic concepts and preliminaries,finally summarizes main results of this thesis.The second chapter is mainly to study the unit graph ofG(Z _n)by using the law of research which is from the paticular to the general.We completely determine the domination number ofG(Z _n)when n has at most three prime divisors.Based on the results in the Chapter 2,in the third chapter,we investigate the domination number of a certain class of finite commutative rings.We completely determine the domination number of G(R)when R is the product of three finite commutative local rings.In Chapter 4,we study the Wiener index and the hyper-Wiener index of unit graph of a finite commutative ring and completely determine the Wiener index and the hyper-Wiener index of unit graph of a finite commutative ring.In Chapter 5,we study the radius of the unit graph of a ring and prove that for a given positive integer n,there exist a ring R such that the radius of G(R)is n.we also prove the radius of the unit graph of right self-injective ring is 1,2,3 and infinite and use the radius to give a classifications of those rings.