Structural Properties of Suborbital Graphs under Product Actions of Wreath Products

This paper examines the suborbital graphs generated by the product actions of the wreath products $S_m \wr S_n$, $A_m \wr S_n$, $C_m \wr S_n$, and $D_{2m} \wr S_n$ on the Cartesian power $X^n$. Using permutation group theory, orbit and stabilizer analysis, combinatorial techniques, and computational verification, the study investigates the classification of suborbits and subdegrees and the structural properties of the resulting graphs. The analysis focuses on the pairing properties of orbitals and graph characteristics, including connectivity, vertex degree, regularity, transitivity, girth, and symmetry. The findings reveal how the orbit structure of each base group, together with coordinate permutations, determines the structure of the associated suborbital graphs. In particular, under suitable conditions, the symmetric and alternating group actions yield adjacency structures governed by Hamming distance, whereas the cyclic and dihedral group actions produce graph structures reflecting their respective permutation actions. The study provides a group-theoretic framework for examining suborbital graphs arising from wreath product actions and contributes to the understanding of the relationship between permutation groups, Cartesian powers, and algebraic graph structures.