Zero-divisor graphs of Catalan monoid

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