The annihilating-ideal graph of commutative semigroups

In this paper, we associate an undirected graph AG(S)AG(S), the annihilating-ideal graph, to a commutative semigroup S  . This graph has vertex set A⁎(S)=A(S)∖{(0)}A⁎(S)=A(S)∖{(0)}, where A(S)A(S) is the set of proper ideals of S   with nonzero annihilator. Two distinct vertices I,J∈A⁎(S)I,J∈A⁎(S) are defined to be adjacent in AG(S)AG(S) if and only if IJ=(0)IJ=(0), the zero ideal. Conditions are given to ensure a finite graph. Semigroups for which each nonzero, proper ideal of S   is an element of A⁎(S)A⁎(S) are characterized. Connections are drawn between AG(S)AG(S) and Γ(S)Γ(S), the well-known zero-divisor graph, and the connectivity, diameter, and girth of AG(S)AG(S) are described. Semigroups S   for which AG(S)AG(S) is a complete or star graph are characterized. Finally, it is proven that the chromatic number is equal to the clique number of the annihilating ideal graph for each reduced semigroup and null semigroup. Upper and lower bounds for χ(AG(S))χ(AG(S)) are given for a general commutative semigroup.