On the annihilator graphs of partial transformation semigroups

Let Formula: see text and Formula: see text be a partial transformation semigroup on Formula: see text Obviously, the empty set Formula: see text is a zero element of Formula: see text and denoted by 0. Let Formula: see text An element Formula: see text is a zero divisor of Formula: see text if there exists Formula: see text such that Formula: see text The set Formula: see text is called the annihilator of Formula: see text in Formula: see text It is clear that Formula: see text for all Formula: see text Let Formula: see text be the set of all zero divisors of Formula: see text and Formula: see text Generally, if Formula: see text then there exists Formula: see text in which Formula: see text In this paper, we construct the annihilator graph Formula: see text of Formula: see text which is an undirected simple graph with vertex set Formula: see text and two distinct vertices Formula: see text and Formula: see text are adjacent if and only if Formula: see text Furthermore, we prove some basic structural properties of Formula: see text and determine invariants for Formula: see text such as the diameter, girth, clique, domination number, and independence number.