Analytical and computational properties of the variable symmetric division deg index

J. A. Méndez-Bermúdez; José M. Rodríguez; José L. Sánchez; José M. Sigarreta; J. A. Méndez-Bermúdez; José M. Rodríguez; José L. Sánchez; José M. Sigarreta

doi:10.3934/mbe.2022413

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 9: 8908-8922. doi: 10.3934/mbe.2022413

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Analytical and computational properties of the variable symmetric division deg index

1.
Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico
2.
Universidad Carlos III de Madrid, Departamento de Matemáticas, Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain
3.
Universidad Autónoma de Guerrero, Centro Acapulco CP 39610, Acapulco de Juárez, Guerrero, Mexico

Academic Editor: Yacine Chitour

Received: 08 May 2022 Revised: 06 June 2022 Accepted: 13 June 2022 Published: 20 June 2022

The aim of this work is to obtain new inequalities for the variable symmetric division deg index $ SDD_\alpha(G) = \sum_{uv \in E(G)} (d_u^\alpha/d_v^\alpha+d_v^\alpha/d_u^\alpha) $, and to characterize graphs extremal with respect to them. Here, by $ uv $ we mean the edge of a graph $ G $ joining the vertices $ u $ and $ v $, and $ d_u $ denotes the degree of $ u $, and $ \alpha \in \mathbb{R} $. Some of these inequalities generalize and improve previous results for the symmetric division deg index. In addition, we computationally apply the $ SDD_\alpha(G) $ index on random graphs and we demonstrate that the ratio $ \langle SDD_\alpha(G) \rangle/n $ ($ n $ is the order of the graph) depends only on the average degree $ \langle d \rangle $.
- symmetric division deg index,
- degree-based topological index,
- random graphs
Citation: J. A. Méndez-Bermúdez, José M. Rodríguez, José L. Sánchez, José M. Sigarreta. Analytical and computational properties of the variable symmetric division deg index[J]. Mathematical Biosciences and Engineering, 2022, 19(9): 8908-8922. doi: 10.3934/mbe.2022413

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Abstract

The aim of this work is to obtain new inequalities for the variable symmetric division deg index $ SDD_\alpha(G) = \sum_{uv \in E(G)} (d_u^\alpha/d_v^\alpha+d_v^\alpha/d_u^\alpha) $, and to characterize graphs extremal with respect to them. Here, by $ uv $ we mean the edge of a graph $ G $ joining the vertices $ u $ and $ v $, and $ d_u $ denotes the degree of $ u $, and $ \alpha \in \mathbb{R} $. Some of these inequalities generalize and improve previous results for the symmetric division deg index. In addition, we computationally apply the $ SDD_\alpha(G) $ index on random graphs and we demonstrate that the ratio $ \langle SDD_\alpha(G) \rangle/n $ ($ n $ is the order of the graph) depends only on the average degree $ \langle d \rangle $.

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