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 GH(R) with all elements of R as the vertices, and two distinct vertices x and y are adjacent if and only if x+y∈H. In this paper, we study some basic properties of GH(R). Moreover, we characterize the planarity, chromatic number, clique number and independence number of GH(R).