Zero-divisor graphs of partial transformation semigroups

Abstract

Let Pn be the partial transformation semigroup on Xn = {1, 2, . . . , n} . In this paper, we find the left zerodivisors, right zero-divisors and two sided zero-divisors of Pn ; and their numbers. For n ≥ 3 , we define an undirected graph Γ(Pn) associated with Pn whose vertices are the two sided zero-divisors of Pn excluding the zero element θ of Pn with distinct two vertices α and β joined by an edge in case αβ = θ = βα . First, we prove that Γ(Pn) is a connected graph, and find the diameter, girth, domination number and the degrees of the all vertices of Γ(Pn) . Furthermore, we give lower bounds for clique number and chromatic number of Γ(Pn) .