On the graph GP(R) over commutative ring R

Let R be a commutative ring with identity 1. Then the graph of R , denoted by G P ( R ) which is defined as the vertices are the elements of R and any two distinct elements a and b are adjacent if and only if the corresponding principal ideals aR and bR satisfy the condition: ( a R ) ( b R ) = a R ⋂ b R . In this paper, we characterize the class of finite commutative rings with 1 for which the graph G P ( R ) is complete. Here we are able to show that the graph G P ( R ) is a line graph of some graph G if and only if G P ( R ) is complete. For n = p 1 r 1 p 2 r 2 … p k r k , we show that chromatic number of G P ( Z n ) is equal to the sum of the number of regular elements in Z n and the number of integers i such that r i > 1 . Moreover, we characterize those n for which the graph G P ( Z n ) is end-regular.