Zero-divisor graphs of Catalan monoid

Abstract

Let C n be the Catalan monoid on X n = {1,. .. , n} under its natural order. In this paper, we describe the sets of left zero-divisors, right zero-divisors and two sided zero-divisors of C n ; and their numbers. For n ≥ 4, we define an undirected graph Γ(C n) associated with C n whose vertices are the two sided zero-divisors of C n excluding the zero element θ of C n with distinct two vertices α and β joined by an edge in case αβ = θ = βα. Then we first prove that Γ(C n) is a connected graph, and then we find the diameter, radius, girth, domination number, clique number and chromatic numbers and the degrees of all vertices of Γ(C n). Moreover, we prove that Γ(C n) is a chordal graph, and so a perfect graph.

Zero-divisor graphs of partial transformation semigroups

“…Furthermore, these definitions can be considered for non-commutative semigroups with zero. Recently, some properties of zero-divisor graphs of Catalan monoids have been researched in [12].…”