Two-sided zero-divisor graphs of orientation-preserving and order-decreasing transformation semigroups - Toker Kemal (Open Mathematics)
For n≥4n\ge 4, let OPDn{{\mathcal{OPD}}}_{n} be the orientation-preserving and order-decreasing transformation semigroup on the finite chain Xn={1<…<n}{X}_{n}=\left\{1\lt … \lt \hspace{0.30em}n\right\}. First, we determine the set of two-sided zero-divisors of OPDn{{\mathcal{OPD}}}_{n}, and its cardinality. Then, we let Γ(OPDn)Γ \left({{\mathcal{OPD}}}_{n}) be the graph whose vertices are the two-sided zero-divisors of OPDn{{\mathcal{OPD}}}_{n} excluding the zero element θθ and distinct two vertices αα and ββ joined by an edge in case αβ=θ=βαα β =θ =β α . In this study, we prove that Γ(OPDn)Γ \left({{\mathcal{OPD}}}_{n}) is a connected graph, and we find the diameter, girth, domination number, minimum degree, and maximum degree of Γ(OPDn)Γ \left({{\mathcal{OPD}}}_{n}). Moreover, we give a lower bound for clique number of Γ(OPDn)Γ \left({{\mathcal{OPD}}}_{n}) and we prove that Γ(OPDn)Γ \left({{\mathcal{OPD}}}_{n}) is an imperfect graph.