On the annihilator graphs of partial transformation semigroups

Abstract

Let X n = { 1 , 2 , … , n } and P n be a partial transformation semigroup on X n . Obviously, the empty set ∅ is a zero element of P n and denoted by 0. Let P n * = P n \\ { 0 } . An element α ∈ P n is a zero divisor of P n if there exists β ∈ P n * such that α β = 0 = β α . The set Ann ( α ) = { β ∈ P n : α β = 0 = β α } is called the annihilator of α in P n . It is clear that 0 ∈ Ann ( α ) for all α ∈ P n . Let Z ( P n ) be the set of all zero divisors of P n and Z ( P n ) * = Z ( P n ) \\ { 0 } . Generally, if α ∈ Z ( P n ) , then there exists β ∈ Z ( P n ) * in which β ∈ Ann ( α ) . In this paper, we construct the annihilator graph Γ n of P n which is an undirected simple graph with vertex set Z ( P n ) * and two distinct vertices α and β are adjacent if and only if Ann ( α ) ∩ Ann ( β ) ≠ { 0 } . Furthermore, we prove some basic structural properties of Γ n and determine invariants for Γ n such as the diameter, girth, clique, domination number, and independence number.