On the divisor graph of finite commutative ring

Abstract

In this paper, we introduce a graphical structure of non empty finite commutative ring R called as divisor graph of R, denoted as D[R], is undirected simple graph with vertex set V=R−{0,1} and for distinct vertices a,b∈V, a∼b if and only if either a|b or b|a, i.e. ∃c∈R such that a=bc or b=ac. We will discuss structure and properties of divisor graph of ring Z_n. Moreover, we also determine diameter, girth, eulerian, planar, clique number of the D[Z_n], ∀n. The main objective of this paper is to study interplay of ring theoretic properties of R with graph theoretic properties of D[Z_n].