Ebrahimi Sarvandi, Z., Ebrahimi Atani, S. (2015). The total graph of a commutative semiring with respect to proper ideals. Journal of Algebra and Related Topics, 3(2), 27-41.

Z. Ebrahimi Sarvandi; S. Ebrahimi Atani. "The total graph of a commutative semiring with respect to proper ideals". Journal of Algebra and Related Topics, 3, 2, 2015, 27-41.

Ebrahimi Sarvandi, Z., Ebrahimi Atani, S. (2015). 'The total graph of a commutative semiring with respect to proper ideals', Journal of Algebra and Related Topics, 3(2), pp. 27-41.

Ebrahimi Sarvandi, Z., Ebrahimi Atani, S. The total graph of a commutative semiring with respect to proper ideals. Journal of Algebra and Related Topics, 2015; 3(2): 27-41.

The total graph of a commutative semiring with respect to proper ideals

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.