Theoretic Properties of \(k^{th}\) Power Graphs of Finite Groups

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In this paper, we investigate the structural and combinatorial properties of the kth power graph Γk(G) associated with a finite group G, where k ≥ 2. The graph Γk(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 = yk or y = xk. 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 Γk(G) is a subgraph of the power graph P(G) and analyze conditions under which Γk(G) is connected, disconnected, or empty. Depending on the algebraic structure of G and the arithmetic properties of k, we show that Γk(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 = Zn and gcd(k, n) = 1, Γk(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 Γk(G) and discuss how subgroup structure and group automorphisms impact the topology of the graph.

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