The Structural Properties of \(\overrightarrow{\mathcal{G}}_0\left(\mathbb{Z}_n\right)\) : Eulerianity, Kernel and Domination Aspects
Jimly Manuel *
Department of Mathematics, Mahatma Gandhi College, Iritty, Kannur, Kerala, India.
Bindhu K Thomas
Department of Mathematics, Mary Matha Arts and Science College, Mananthavady, India.
*Author to whom correspondence should be addressed.
Abstract
In this research paper, we undertake a detailed study of a subdigraph of \(\overrightarrow{\mathcal{G}}_0\left(\mathbb{Z}_n\right)\) for n ≥ 2, formed by removing the vertex 0. This resulting subdigraph is denoted by \(\overrightarrow{\mathcal{G}}_0\left(\mathbb{Z}_n\right)\) We establish the necessary and sufficient conditions under which \(\overrightarrow{\mathcal{G}}_0\left(\mathbb{Z}_n\right)\) is Eulerian. Furthermore, we investigate the kernel of this digraph and analyze domination and twin domination properties within the framework of \(\overrightarrow{\mathcal{G}}_0\left(\mathbb{Z}_n\right)\).
Keywords: Directed power graph \(\overrightarrow{\mathcal{G}}_0\left(\mathbb{Z}_n\right)\) , \(\overrightarrow{\mathcal{G}}_0\left(\mathbb{Z}_n\right)\) , eulerian graph, kernel, domination, twin dominating number