Mean Sombor index

TL;DR

The paper introduces the mean Sombor index, a new degree-based topological index inspired by the power mean, which relates to known indices and correlates with physicochemical properties of octane isomers.

Contribution

It presents the definition, mathematical properties, bounds, and relationships of the mean Sombor index, a novel index inspired by the power mean and related to various known topological indices.

Findings

The mean Sombor index generalizes several known indices.

It shows strong correlation with physicochemical properties of octane isomers.

Mathematical bounds and relationships with existing indices are established.

Abstract

We introduce a degree-based variable topological index inspired on the power (or generalized) mean. We name this new index as the mean Sombor index: $mSO_\alpha(G) = \sum_{uv \in E(G)} \left[\left( d_u^\alpha+d_v^\alpha \right) /2 \right]^{1/\alpha}$. Here, $uv$ denotes the edge of the graph $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $\alpha \in \mathbb{R} \backslash \{0\}$. We also consider the limit cases $mSO_{\alpha\to 0}(G)$ and $mSO_{\alpha\to\pm\infty}(G)$. Indeed, for given values of $\alpha$, the mean Sombor index is related to well-known topological indices such as the inverse sum indeg index, the reciprocal Randic index, the first Zagreb index, the Stolarsky--Puebla index and several Sombor indices. Moreover, through a quantitative structure property relationship (QSPR) analysis we show that $mSO_\alpha(G)$ correlates well with several physicochemical properties of octane isomers. Some mathematical properties of mean Sombor indices as well as bounds and new relationships with known topological indices are also discussed.