Esmaeili Khalil Saraei, F., Navidinia, E. (2018). A note on the extended total graph of commutative rings. Journal of Algebra and Related Topics, 6(1), 25-33. doi: 10.22124/jart.2018.10241.1101

F. Esmaeili Khalil Saraei; E. Navidinia. "A note on the extended total graph of commutative rings". Journal of Algebra and Related Topics, 6, 1, 2018, 25-33. doi: 10.22124/jart.2018.10241.1101

Esmaeili Khalil Saraei, F., Navidinia, E. (2018). 'A note on the extended total graph of commutative rings', Journal of Algebra and Related Topics, 6(1), pp. 25-33. doi: 10.22124/jart.2018.10241.1101

Esmaeili Khalil Saraei, F., Navidinia, E. A note on the extended total graph of commutative rings. Journal of Algebra and Related Topics, 2018; 6(1): 25-33. doi: 10.22124/jart.2018.10241.1101

A note on the extended total graph of commutative rings

^{2}Department of Mathematics, University of Guilan, Rasht, Iran

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 a 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.