TY - JOUR
ID - 1539
TI - The total graph of a commutative semiring with respect to proper ideals
JO - Journal of Algebra and Related Topics
JA - JART
LA - en
SN - 2345-3931
AU - Ebrahimi Sarvandi, Z.
AU - Ebrahimi Atani, S.
AD - University of Guilan
Y1 - 2015
PY - 2015
VL - 3
IS - 2
SP - 27
EP - 41
KW - Commutative semirings
KW - Zero-divisor
KW - Total graph
DO -
N2 - 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.
UR - https://jart.guilan.ac.ir/article_1539.html
L1 - https://jart.guilan.ac.ir/article_1539_c4cf79602c757856bbf5ef810db8ebf5.pdf
ER -