Stolarsky-Puebla index

TL;DR

The paper introduces the Stolarsky-Puebla index, a new degree-based topological index inspired by the Stolarsky mean, unifying several known indices and analyzing their behavior on random networks.

Contribution

A novel topological index based on Stolarsky mean is proposed, generalizing and connecting existing indices, with applications to random network analysis.

Findings

The index reproduces known indices like the reciprocal Randic and Zagreb indices.

The average index scales with the network's average degree.

Application to random networks demonstrates the index's behavior.

Abstract

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 $\left< d \right>$.