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Revan Sombor indices: Analytical and statistical study.

V R Kulli1, J A Méndez-Bermúdez2, José M Rodríguez3

  • 1Department of Mathematics, Gulbarga University, Gulbarga 585106, India.

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Summary

This study analyzes Revan indices, including the Revan Sombor index, on graphs. New relations bound these indices and connect them to other graph invariants for statistical analysis of random graphs.

Keywords:
(a,b)-KA indicesRevan Sombor indexdegree-based topological indicesmodified Sombor indexrandom graph

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Area of Science:

  • Graph theory
  • Chemical graph theory
  • Combinatorial mathematics

Background:

  • Revan indices are graph invariants derived from vertex degrees.
  • The Revan degree of a vertex $r_u$ is defined as $\Delta + \delta - d_u$, where $\Delta$ and $\delta$ are the maximum and minimum degrees in the graph, and $d_u$ is the degree of vertex $u$.
  • This work focuses on Revan indices within the Sombor family, specifically the Revan Sombor index and the first and second Revan $(a, b)$-KA indices.

Purpose of the Study:

  • To analytically and statistically study Revan indices on graphs.
  • To establish new relations for bounding Revan Sombor indices.
  • To extend these relations for the statistical study of random graph ensembles.

Main Methods:

  • Analytical and statistical studies of graph indices.
  • Derivation of new relations for bounds on Revan Sombor indices.
  • Extension of these relations to average index values for statistical applications.

Main Results:

  • New relations are presented for bounding Revan Sombor indices.
  • These bounds connect Revan Sombor indices with other Revan indices (e.g., Revan Zagreb indices) and standard degree-based indices (e.g., Sombor index, Zagreb index, Harmonic index).
  • The study extends relations to average index values, facilitating statistical analysis of random graphs.

Conclusions:

  • The paper provides novel insights into the properties and applications of Revan indices, particularly the Revan Sombor index.
  • The established relations enhance the understanding of graph structure and relationships between different graph invariants.
  • The findings support the use of Revan indices in the statistical analysis of complex network structures.