%0 Journal Article
%T A note on the extended total graph of commutative rings
%J Journal of Algebra and Related Topics
%I University of Guilan
%Z 2345-3931
%A Esmaeili Khalil Saraei, F.
%A Navidinia, E.
%D 2018
%\ 06/01/2018
%V 6
%N 1
%P 25-33
%! A note on the extended total graph of commutative rings
%K Total graph
%K prime ideal
%K multiplicative-prime subset
%R 10.22124/jart.2018.10241.1101
%X 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.
%U https://jart.guilan.ac.ir/article_3024_c7a72ce759419a2bcd9ddd57dcf4da67.pdf