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.
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