A note on the extended total graph of commutative rings

Let R be a commutative ring and H a nonempty proper subset of R.In this paper, the extended total graph, denoted by ET_{H}(R) is presented, where H is amultiplicative-prime subset of R. It is the graph with all elements of R as vertices, and for distinct p,q\in R, the vertices p and q are adjacent if and only if rp+sq\in H for some r,s\in R\setminus H. We also study the two (induced) subgraphs ET_{H}(H) and ET_{H}(R\setminus H), with vertices H and R\setminus H, respectively. Among other things, the diameter and the girth of ET_{H}(R) are also studied.

Keywords:

Total graph , prime ideal , multiplicative-prime subset