Abstract:Let $I$ be a strong co-ideal of a commutative semiring $R$ with identity. Let $\Gamma_{I} (R)$ be a graph with the set of vertices $S_{I} (R) = \{x \in R\setminus I: ~ x + y \in I$ for ~ some $y \in R \setminus I\}$, where two distinct vertices $x$ and $y$ are adjacent if and only if $x + y \in I$. We look at the diameter and girth of this graph. Also we discuss when $\Gamma_{I} (R)$ is bipartite. Moreover, studies are done on the planarity, clique, and chromatic number of this graph. Examples illustrating the results are presented.
Keywords: strong co-ideal; $Q$-strong co-ideal; identity-summand element; identity-summand graph; co-ideal based
DOI: DOI 10.14712/1213-7243.2015.124
AMS Subject Classification: 16Y60 05C62