Revan Sombor indices: Analytical and statistical study

V. R. Kulli; J. A. Méndez-Bermúdez; José M. Rodríguez; José M. Sigarreta; V. R. Kulli; J. A. Méndez-Bermúdez; José M. Rodríguez; José M. Sigarreta

doi:10.3934/mbe.2023082

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 2: 1801-1819. doi: 10.3934/mbe.2023082

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Revan Sombor indices: Analytical and statistical study

1.
Department of Mathematics, Gulbarga University, Gulbarga 585106, India
2.
Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico
3.
Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain
4.
Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame No.54 Col. Garita, 39650 Acalpulco Gro., Mexico

Academic Editor: Yang Kuang

Received: 27 August 2022 Revised: 11 October 2022 Accepted: 19 October 2022 Published: 07 November 2022

In this paper, we perform analytical and statistical studies of Revan indices on graphs $ G $: $ R(G) = \sum_{uv \in E(G)} F(r_u, r_v) $, where $ uv $ denotes the edge of $ G $ connecting the vertices $ u $ and $ v $, $ r_u $ is the Revan degree of the vertex $ u $, and $ F $ is a function of the Revan vertex degrees. Here, $ r_u = \Delta + \delta - d_u $ with $ \Delta $ and $ \delta $ the maximum and minimum degrees among the vertices of $ G $ and $ d_u $ is the degree of the vertex $ u $. We concentrate on Revan indices of the Sombor family, i.e., the Revan Sombor index and the first and second Revan $ (a, b) $-$ KA $ indices. First, we present new relations to provide bounds on Revan Sombor indices which also relate them with other Revan indices (such as the Revan versions of the first and second Zagreb indices) and with standard degree-based indices (such as the Sombor index, the first and second $ (a, b) $-$ KA $ indices, the first Zagreb index and the Harmonic index). Then, we extend some relations to index average values, so they can be effectively used for the statistical study of ensembles of random graphs.
- Revan Sombor index,
- modified Sombor index,
- $ (a, b) $-$ KA $ indices,
- degree-based topological indices,
- random graph
Citation: V. R. Kulli, J. A. Méndez-Bermúdez, José M. Rodríguez, José M. Sigarreta. Revan Sombor indices: Analytical and statistical study[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 1801-1819. doi: 10.3934/mbe.2023082

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Abstract

In this paper, we perform analytical and statistical studies of Revan indices on graphs $ G $: $ R(G) = \sum_{uv \in E(G)} F(r_u, r_v) $, where $ uv $ denotes the edge of $ G $ connecting the vertices $ u $ and $ v $, $ r_u $ is the Revan degree of the vertex $ u $, and $ F $ is a function of the Revan vertex degrees. Here, $ r_u = \Delta + \delta - d_u $ with $ \Delta $ and $ \delta $ the maximum and minimum degrees among the vertices of $ G $ and $ d_u $ is the degree of the vertex $ u $. We concentrate on Revan indices of the Sombor family, i.e., the Revan Sombor index and the first and second Revan $ (a, b) $-$ KA $ indices. First, we present new relations to provide bounds on Revan Sombor indices which also relate them with other Revan indices (such as the Revan versions of the first and second Zagreb indices) and with standard degree-based indices (such as the Sombor index, the first and second $ (a, b) $-$ KA $ indices, the first Zagreb index and the Harmonic index). Then, we extend some relations to index average values, so they can be effectively used for the statistical study of ensembles of random graphs.

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