On the Orbital Regular Graph of Finite Solvable Groups

Let G be a finite solvable group and Δ be the subset of Υ×Υ, where Υ is the set of all pairs of size two commuting elements in G. If G operates on a transitive G – space by the action (υ1,υ2)g=(υg1,υg2); υ1,υ2∈Υ and g∈G, then orbits of G are called orbitals. The subset Δo={(υ,υ);υ∈Υ,(υ,υ)∈Υ×Υ} represents G′s diagonal orbital.The orbital regular graph is a graph on which G acts regularly on the vertices and the edge set. In this paper, we obtain the orbital regular graphs for some finite solvable groups using a regular action. Furthermore, the number of edges for each of a group’s orbitals is obtained.