Theoretic Properties of k^th Power Graphs of Finite Groups

In this paper, we investigate the structural and combinatorial properties of the kth power graph Γ_k (G) associated with a finite group G, where k ≥ 2. The graph Γ_k (G) is defined by taking the elements of G as vertices and connecting two distinct vertices x and y by an edge if either x = y^k or y = x^k . This construction generalizes the well-studied power graph of a group and provides new insight into the influence of exponentiation on group elements when viewed through graph-theoretical properties. We show that Γ_k (G) is a subgraph of the power graph P(G) and analyze conditions under which Γ_k (G) is connected, disconnected, or empty. Depending on the algebraic structure of G and the arithmetic properties of k, we show that Γ_k (G) can exhibit a variety of structural forms, including being a tree, a union of disjoint stars, or a complete multipartite graph. For instance, when G = Z_n and gcd(k, n) = 1, Γ_k (G) decomposes into disjoint stars, while for certain non-cyclic groups, the graph becomes multipartite. Additionally, we provide formulas for computing the number of edges in Γ_k (G) and discuss how subgroup structure and group automorphisms impact the topology of the graph.