On the weakly nilpotent graph of a commutative semiring

Abstract

Let S be a commutative semiring with unity. In this paper, we introduce the weakly nilpotent graph of a commutative semiring. The weakly nilpotent graph of S, denoted by 􀀀w(S) is defined as an undirected simple graph whose vertices are S and two distinct vertices x and y are adjacent if and only if xy 2 N(S), where S = Sn f0g and N(S) is the set of all non-zero nilpotent elements of S. In this paper, we determine the diameter of weakly nilpotent graph of an Artinian semiring. We prove that if 􀀀w(S) is a forest, then 􀀀w(S) is a union of a star and some isolated vertices. We study the clique number, the chromatic number and the independence number of 􀀀w(S). Among other results, we show that for an Artinian semiring S, 􀀀w(S) is not a disjoint union of cycles or a unicyclic graph. For Artinian semirings, we determine diam(􀀀w(S)). Finally, we characterize all commutative semirings S for which 􀀀w(S) is a cycle, where 􀀀w(S) is the complement of the weakly nilpotent graph of S. Finally, we characterize all commutative semirings S for which Γw(S) is a cycle.