Multiplicative topological indices: Analytical properties and application to random networks

R. Aguilar-Sánchez; J. A. Mendez-Bermudez; José M. Rodríguez; José M. Sigarreta; R. Aguilar-Sánchez; J. A. Mendez-Bermudez; José M. Rodríguez; José M. Sigarreta

doi:10.3934/math.2024179

AIMS Mathematics

2024, Volume 9, Issue 2: 3646-3670. doi: 10.3934/math.2024179

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Multiplicative topological indices: Analytical properties and application to random networks

1.
Facultad de Ciencias Químicas, Benemérita Universidad Autónoma de Puebla, Puebla 72570, Mexico
2.
Instituto de Física, Benemérita Universidad Autónoma de Puebla, Puebla 72570, Mexico
3.
Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, Leganés, Madrid 28911, Spain
4.
Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame No. 54 Col. Garita, Acapulco, Guerrero 610101, Mexico

Received: 20 September 2023 Revised: 21 December 2023 Accepted: 25 December 2023 Published: 09 January 2024
MSC : 05C50, 05C80, 05C92, 60B20

We consider two general classes of multiplicative degree-based topological indices (MTIs), denoted by $ X_{\Pi, F_V}(G) = \prod_{u \in V(G)} F_V(d_u) $ and $ X_{\Pi, F_E}(G) = \prod_{uv \in E(G)} F_E(d_u, d_v) $, where $ uv $ indicates the edge of $ G $ connecting the vertices $ u $ and $ v $, $ d_u $ is the degree of the vertex $ u $, and $ F_V(x) $ and $ F_E(x, y) $ are functions of the vertex degrees. This work has three objectives: First, we follow an analytical approach to deal with a classical topic in the study of topological indices: to find inequalities that relate two MTIs between them, but also to their additive versions $ X_\Sigma(G) $. Second, we propose some statistical analysis of MTIs as a generic tool for studying average properties of random networks, extending these techniques for the first time to the context of MTIs. Finally, we perform an innovative scaling analysis of MTIs which allows us to state a scaling law that relates different random graph models.
- multiplicative topological index,
- degree-based topological index,
- random graphs
Citation: R. Aguilar-Sánchez, J. A. Mendez-Bermudez, José M. Rodríguez, José M. Sigarreta. Multiplicative topological indices: Analytical properties and application to random networks[J]. AIMS Mathematics, 2024, 9(2): 3646-3670. doi: 10.3934/math.2024179

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Abstract

We consider two general classes of multiplicative degree-based topological indices (MTIs), denoted by $ X_{\Pi, F_V}(G) = \prod_{u \in V(G)} F_V(d_u) $ and $ X_{\Pi, F_E}(G) = \prod_{uv \in E(G)} F_E(d_u, d_v) $, where $ uv $ indicates the edge of $ G $ connecting the vertices $ u $ and $ v $, $ d_u $ is the degree of the vertex $ u $, and $ F_V(x) $ and $ F_E(x, y) $ are functions of the vertex degrees. This work has three objectives: First, we follow an analytical approach to deal with a classical topic in the study of topological indices: to find inequalities that relate two MTIs between them, but also to their additive versions $ X_\Sigma(G) $. Second, we propose some statistical analysis of MTIs as a generic tool for studying average properties of random networks, extending these techniques for the first time to the context of MTIs. Finally, we perform an innovative scaling analysis of MTIs which allows us to state a scaling law that relates different random graph models.

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