Annihilator graph of a commutative semigroup whose zero divisor graph is a star graph [Articol]

Abstract

Let S be a commutative semigroup with zero and Г(S) be the zero divisor graph of S which is a star graph with center x and n end vertices. Also, let Z (S) be the set of all nonzero zero divisors of S. The annihilator graph for a commutative semigroup S, which is denoted by AG(S), is an undirected graph with vertex set Z (S) and two distinct vertices a and b are adjacent if and only if annS(ab) 6= annS(a) [ annS(b), where annS(a) = fs 2 S j as = 0g. In this paper, we study the annihilator graph associated with a commutative semigroup with zero, when Г(S) is a star graph. We prove that if Z(S) 6= S, then AG(S) is a star graph or a complete graph, and if If Z(S) = S, then AG(S) is an empty graph or AG(S) = K1 [ Kn. Moreover, we show that if AG(S) is a star graph with center x, then Z(S) 6= S and Г(S) is a star graph with center x and the product of all end vertices of Г(S) is not equal to the center vertex.