Structures Associated with the Direct Product of Alternating and Dihedral Groups Acting on the Cartesian Product of Two Sets

Abstract

Whenever a permutation group acts on a set, combinatorial and invariant properties, and mathematical structures that result from this group action are studied. Various mathematicians have studied these properties over time using different groups acting on both ordered and unordered sets. The combinatorial properties (transitivity and primitivity) and invariants (ranks and subdegrees) of the direct product between alternating and dihedral groups acting on the Cartesian product of two sets have already been studied and it was found out that the group action is transitive, imprimitive, therankis6, and sub degrees are obtained according to theorem 2.3. This research seeks to extend this by constructing and analyzing the properties (simple/multigraph, self-pairedness, connectedness, degree of the vertex, girth, and directedness) of these mathematical structures (suborbital graphs) that result from the group action. This research for n ≥ 3, suborbital graphs can be classified into three categories; First, those constructed when only the first components of the vertex set are identical and second, those when only the second components of the vertex set are identical. The suborbital graphs of the first and second category are simple, self-paired, have n− disconnected components, are regular with degree n − 1 and girth is 3. The third category of suborbital graphs in which neither the first nor the second components of the vertex set are identical and they are; simple, self-paired, connected, regular with degree of vertex varying from graph to graph, and girth 3.