Two-Sided Zero-Divisor Graphs of Order-Preserving and A-Decreasing Finite Transformation Semigroups

Let On(A) be the order-preserving and A-decreasing finite transformation semigroup on Xn={1,2,…,n}. It is known that On(A) has zero element if and only if 1∈A. In this paper, we investigate zero-divisor graphs of On(A) where 1∈A and n≥4. First, we determine the set of right, left, and two-sided zero-divisors of On(A); and their cardinalities. Let Γ(On(A)) be the undirected graph whose vertices are the two sided zero-divisors of On(A) excluding the zero element (θ) and distinct two vertices α and β joined by an edge if and only if αβ=θ=βα. In this paper, we prove that Γ(On(A)) is a connected graph and find its diameter, girth, domination number and degrees of all vertices in Γ(On(A)). Moreover, we prove that Γ(On(A)) is a perfect graph and we calculate clique number and chromatic number of it.