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∈R∖H. 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