On the annihilator graphs of partial transformation semigroups

Abstract

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.

Keywords

• Annihilator graph
• diameter
• domination number
• girth
• partial transformation semigroup
• zero divisor