On Graphs Defined on Algebraic Objects

Abstract

Starting from Cayley graphs, the research on graphs defined on algebraic objects like groups, rings, vector spaces forms an integral part of algebraic graph theory. Such graphs reflect the algebraic properties of the underlying structure and hence it is useful to study the algebraic objects with the help of graphs defined on them. Our present study is focused on defining graphs on groups and rings and investigate their properties. This includes studying the interplay between algebraic properties of the objects and their corresponding graph properties. The graphs studied by us are defined below: LetGbe a group andSbe the collection of all non-trivial proper subgroups ofG. The comaximal subgroup graph ofG, denoted byΓ(G), whose vertex set is Sand two vertices HandKare adjacent if and only ifHK=G. LetRbe a commutative ring with identity. The prime ideal sum graph ofR, denoted byG(R), is a graph whose vertices are non-zero proper ideals ofRand two distinct vertices IandJare adjacent if and only ifI+Jis a prime ideal ofR. We study different structural and isomorphism properties of these graphs which pro- vides nice insights on the algebraic as well as graph theoretic aspects.