Annihilator graph of semiring of matrices over Boolean semiring

Abstract

Abstract The annihilator graph of a semiring S, denoted by AG(S), is the graph whose vertex set is the set of all nonzero zero-divisors of S. In commutative semiring S, two distinct vertices are adjacent if and only if ann(xy) ≠ ann(x) ∪ ann(y), where ann(x) = {s ∈ S|sx = 0}. Similarly in noncommutative semiring S, two distinct vertices are connected by an edge if and only if either l. ann(xy) ≠ l. ann(x) ∪ l. ann(y), l. ann(yx) ≠ l. ann(x) ∪ l. ann(y), r. ann(xy) ≠ r. ann(x) ∪ r. ann(y), or r. ann(yx) ≠ r. ann(x) ∪ r. ann(y) where l. ann(x) = {s ∈ S|sx = 0} and r. ann(x) = {s ∈ S|xs = 0}. In this paper we study the properties of the right annihilator and the left annihilator of semiring of matrices over Boolean semiring Mn (ℬ) and then use these results to determine the diameter of the graph AG(Mn (ℬ)).