Theoretic Properties of \(k^{th}\) Power Graphs of Finite Groups
Bibliographic record
Abstract
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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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".