Suborbital Graphs Involving the Action of the Direct Product of Alternating and Cyclic Groups on the Cartesian Product of Two Sets

The combinatorial properties and invariants which includes transitivity, primitivity, ranks and subdegrees of direct product of Alternating group and Cyclic group acting on Cartesian product of two set have been extensively studied. However, the construction of suborbital graphs for this group action remains largely unexplored. As a result, this research paper addresses this gap by constructing suborbital graphs involving direct product of the Alternating group and Cyclic group acting on the Cartesian product of two sets where their respective properties such as connectivity, self-pairedness, girth and vertex degree are analyzed in detail. First, the result shows that all suborbits are <i>self − paired</i> implies that the vertex sets are undirected to each other. Secondly, the constructed suborbital graphs are classified into three parts for any value of <i>n</i> ≥ 3. In <i>part A</i>, it is proven that the constructed suborbital graphs Γ<sub>1</sub>, Γ<sub>2</sub>, Γ<sub>3</sub>,..., Γ<sub>(<i>n</i>−1)</sub> are undirected, regular of degree (<i>n</i>−1), disconnected with <i>n−connected</i> components and has a girth of 3. In <i>part B</i>, the constructed suborbital graph Γ<sub>(n−1)+1</sub> is found to be undirected, regular of degree (n − 1), disconnected with n − connected components and has a girth of 3. Lastly in <i>part C</i>, the graphs Γ<sub>(n−1)+2</sub>, Γ<sub>(n−1)+3</sub>, Γ<sub>(n−1)+4</sub>,..., Γ<sub>(n−1)+n</sub> are found to be undirected, regular of degree (<i>n</i> − 1)<sup>2</sup>, disconnected with <i>n</i><sup>2</sup>− <i>connected</i> components and has a girth of 3.