Comparison of two methods for calculating ranking points using transitive triads

Document Type : Full Length Article

Author

Department of Mathematics, Shenzhen MSU-BIT University, Shenzhen, China.

Abstract

To date, the feasibility of constructing a complete graph invariant in polynomial time remains uncertain. Therefore, developing fast algorithms for checking non-isomorphism, including heuristic ones, is crucial. Successful implementation of these heuristics involves modifying existing graph invariants and creating new ones, both of which are still pertinent. Many existing invariants enable the distinction of a large number of graphs in real time.
This paper introduces an invariant specifically for tournaments, a type of directed graph. Tournaments are interesting because the number of different tournaments, given a fixed order of vertices, matches the number of undirected graphs with the same fixed order. The proposed invariant considers all possible tournaments formed by subsets of vertices from the given digraph with the same set of arcs. For each subset tournament, standard places are calculated and summed to determine the final vertex points, which constitute the new invariant.
Our calculations reveal that the new invariant differs from the most natural tournament invariant, which assigns points to each participant. Initial computational experiments show that the smallest pair correlation between sequences representing these two invariants is observed at dimension 15.

Graphical Abstract

Comparison of two methods for calculating ranking points using transitive triads

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References

[1] D. Pacheco, L. de Lima, C.S. Oliveira, On the Graovac–Ghorbani index for bicyclic
graphs with no pendent vertices, MATCH Commun. Math. Comput. Chem. 86 (2021) 429–
448.https://doi.org/10.48550/arXiv.2005.02141
[2] L. Beretta, F. M. Nardini, R. Trani, R. Venturini, An Optimal Algorithm for Finding Champions in
Tournament Graphs, arXiv, 2018. https://doi.org/10.1109/tkde.2023.3267345
[3] J. Shen, L. Sheng, J. Wu, Searching for Sorted Sequences of Kings in Tournaments, SIAM J. Comput.
32 (2003) 1201-1209. https://doi.org/10.1137/S0097539702410053
[4] R. Gera, T. W. Haynes, S. T. Hedetniemi, Graph Theory: Favorite Conjectures and Open Problems
- 2, Springer, 2018. https://doi.org/10.1007/978-3-319-97686-0
[5] B. F. Melnikov, B. Liu, On an Invariant of Tournament Digraphs, J. Appl. Math. Phys. 12 (2024)
2711–2722. https://doi.org/10.4236/jamp.2024.127162
Journal of Discrete Mathematics and Its Applications
Volume 9, Issue 4
December 2024
Pages 309-320

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History

  • Receive Date: 17 October 2024
  • Revise Date: 19 November 2024
  • Accept Date: 25 November 2024
  • Publish Date: 01 December 2024

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