Stolarsky-Puebla index

We introduce a degree-based variable topological index inspired on the Stolarsky mean (known as the generalization of the logarithmic mean). We name this new index as the Stolarsky-Puebla index: \(SP_\alpha(G) = \sum_{uv \in E(G)} d_u\), if \(d_u=d_v\), and \(SP_\alpha(G) = \sum_{uv \in E(G)} \left[...

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Bibliographic Details
Published in:arXiv.org 2021-09
Main Authors: Mendez-Bermudez, J A, Aguilar-Sanchez, R, Ricardo Abreu Blaya, Sigarreta, Jose M
Format: Article
Language:English
Subjects:
Apexes
Graph theory
Topology
Online Access:Get full text
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Summary:We introduce a degree-based variable topological index inspired on the Stolarsky mean (known as the generalization of the logarithmic mean). We name this new index as the Stolarsky-Puebla index: \(SP_\alpha(G) = \sum_{uv \in E(G)} d_u\), if \(d_u=d_v\), and \(SP_\alpha(G) = \sum_{uv \in E(G)} \left[\left( d_u^\alpha-d_v^\alpha\right)/\left( \alpha(d_u-d_v\right)\right]^{1/(\alpha-1)}\), otherwise. Here, \(uv\) denotes the edge of the network \(G\) connecting the vertices \(u\) and \(v\), \(d_u\) is the degree of the vertex \(u\), and \(\alpha \in \mathbb{R} \backslash \{0,1\}\). Indeed, for given values of \(\alpha\), the Stolarsky-Puebla index reproduces well-known topological indices such as the reciprocal Randic index, the first Zagreb index, and several mean Sombor indices. Moreover, we apply these indices to random networks and demonstrate that \(\left< SP_\alpha(G) \right>\), normalized to the order of the network, scale with the corresponding average degree \(\left< d \right>\).
ISSN:2331-8422