Co-annihilating graphs of partial transformation semigroups: structure and basic invariants

Let [Formula: see text] and [Formula: see text] Denote by [Formula: see text] the partial transformation semigroup on [Formula: see text] Clearly, the empty set [Formula: see text] is a zero element of [Formula: see text] which is written as 0. Let [Formula: see text] An element [Formula: see text] is called a zero-divisor of [Formula: see text] if there exists [Formula: see text] in which [Formula: see text] The annihilator [Formula: see text] of [Formula: see text] is defined to be the set [Formula: see text] Obviously, [Formula: see text] always contains 0. Further, denote by [Formula: see text] the set of all zero-divisors of [Formula: see text] and [Formula: see text] In this paper, the co-annihilating graph [Formula: see text] of [Formula: see text] is constructed as an undirected simple graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are joined by edge if and only if [Formula: see text] Certain properties of [Formula: see text] are investigated. Moreover, some basic invariants of [Formula: see text] are also determined.