Stolarsky-Puebla index

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 $\left< d \right>$.