Zero-divisor graphs of Catalan monoid

Abstract

Let $\mathcal C_{n}$ be the Catalan monoid on $X_{n}=\{1,\ldots ,n\}$ under its natural order. In this paper, we describe the sets of left zero-divisors, right zero-divisors and two sided zero-divisors of $\mathcal C_{n}$; and their numbers. For $n \geq 4$, we define an undirected graph $\Gamma(\mathcal C_{n})$ associated with $\mathcal C_{n}$ whose vertices are the two sided zero-divisors of $\mathcal C_{n}$ excluding the zero element $\theta$ of $\mathcal C_{n}$ with distinct two vertices $\alpha$ and $\beta$ joined by an edge in case $\alpha\beta=\theta=\beta\alpha$. Then we first prove that $\Gamma(\mathcal C_{n})$ 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 $\Gamma(\mathcal C_{n})$. Moreover, we prove that $\Gamma(\mathcal C_{n})$ is a chordal graph, and so a perfect graph.