A graph associated to a commutative semiring

Abstract

Let $R$ be a commutative finite semiring with nonzero identity and $H$ be an arbitrary multiplicatively closed subset $R$. The generalized identity-summand graph of $R$ is the (simple) graph $G_H\left(R\right)$ with all elements of $R$ as the vertices, and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in H$. In this paper, we study some basic properties of $G_H\left(R\right)$. Moreover, we characterize the planarity, chromatic number, clique number and independence number of $G_H\left(R\right)$.

Keywords

• I-semiring
• planar graphs
• clique number
• chromatic number
• independence number

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