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).