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 ETH(R) is presented, where H is a multiplicative-prime subset of R. It is the graph with all elements of R as vertices, and for distinct p, q ∈ R, the vertices p and q are adjacent if and only if rp+sq ∈ H for some r, s ∈ RH. We also study the two (induced) subgraphs ETH(H) and ETH(R H), with vertices H and R H, respectively. Among other things, the diameter and the girth of ETH(R) are also studied.