University of Guilan Journal of Algebra and Related Topics 2345-3931 6 1 2018 06 01 A note on the extended total graph of commutative rings 25 33 3024 10.22124/jart.2018.10241.1101 EN F. Esmaeili Khalil Saraei University of Tehran E. Navidinia Department of Mathematics, University of Guilan, Rasht, Iran Journal Article 2018 04 25 Let $R$ be a commutative ring and $H$ a nonempty proper subset of $R$.<br />In this paper, the extended total graph, denoted by $ET_{H}(R)$ is presented, where $H$ is a<br />multiplicative-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. https://jart.guilan.ac.ir/article_3024_c7a72ce759419a2bcd9ddd57dcf4da67.pdf