A note on the extended total graph of commutative rings

Abstract

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 $p,qin R$, the vertices $p$ and $q$ are adjacent if and only if $rp+sqin H$ for some $r,sin Rsetminus H$. We also study the two (induced) subgraphs $ET_{H}(H)$ and $ET_{H}(Rsetminus H)$, with vertices $H$ and $Rsetminus H$, respectively. Among other things, the diameter and the girth of $ET_{H}(R)$ are also studied.