%0 Journal Article
%T The total graph of a commutative semiring with respect to proper ideals
%J Journal of Algebra and Related Topics
%I University of Guilan
%Z 2345-3931
%A Ebrahimi Sarvandi, Z.
%A Ebrahimi Atani, S.
%D 2015
%\ 12/01/2015
%V 3
%N 2
%P 27-41
%! The total graph of a commutative semiring with respect to proper ideals
%K Commutative semirings
%K Zero-divisor
%K Total graph
%R
%X Let $I$ be a proper ideal of a commutative semiring $R$ and let $P(I)$ be the set of all elements of $R$ that are not prime to $I$. In this paper, we investigate the total graph of $R$ with respect to $I$, denoted by $T(\Gamma_{I} (R))$. It is the (undirected) graph with elements of $R$ as vertices, and for distinct $x, y \in R$, the vertices $x$ and $y$ are adjacent if and only if $x + y \in P(I)$. The properties and possible structures of the two (induced) subgraphs $P(\Gamma_{I} (R))$ and $\bar {P}(\Gamma_{I} (R))$ of $T(\Gamma_{I} (R))$, with vertices $P(I)$ and $R - P(I)$, respectively are studied.
%U https://jart.guilan.ac.ir/article_1539_c4cf79602c757856bbf5ef810db8ebf5.pdf