Zero-divisor graphs of Catalan monoid

Authors : Kemal TOKER

Pages : 387-396

Doi:10.15672/hujms.702478

View : 18 | Download : 7

Publication Date : 2021-04-11

Article Type : Research Paper

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 $\Gammainsert ignore into journalissuearticles values(\mathcal C_{n});$ 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 $\Gammainsert ignore into journalissuearticles values(\mathcal C_{n});$ 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 $\Gammainsert ignore into journalissuearticles values(\mathcal C_{n});$. Moreover, we prove that $\Gammainsert ignore into journalissuearticles values(\mathcal C_{n});$ is a chordal graph, and so a perfect graph.
Keywords : Catalan monoid, zero divisor graph, perfect graph, clique number