Zero-divisor graphs of noncommutative semigroups

In this thesis, zero divisor graphs of order-decreasing full transformation semigroups are investigated. Let n∈Z^+ and X_n={1,2,… ,n} be a finite set. Let D_n be the order-decreasing full transformation semigroup on X_n. We find the left zero-divisors, the right zero-divisors and two sided zero divisors of D_n. Moreover, for n≥4 we define an undirected graph Γ(D_n) whose vertices are two-sided zero divisors of D_n excluding the zero element θ of D_n. In the graph, distinct two vertices α and β are adjacent if and only if αβ=θ=βα. We prove that Γ(D_n) is a connected graph, and we find diameter, girth, the degrees of all vertices, the maximum degree and the minimum degree in Γ(D_n). Moreover, we give lower bounds for clique number and choromatic number of Γ(D_n).