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: $S{P}_{\alpha }(G)={\sum }_{uv\in E(G)}{d}_u$, if $d_u=d_v$, and $S{P}_{\alpha }(G)={\sum }_{uv\in E(G)}{\left[\left(d_u^{\alpha }-d_v^{\alpha }\right)/\left(\alpha (d_u-d_v\right)\right]}^{1/(\alpha -1)}$, 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 $⟨S{P}_{\alpha }(G)⟩$, normalized to the order of the network, scale with the corresponding average degree $⟨d⟩$.

7 pages, 4 figures, accepted for publication in the special volume of Discrete Mathematics Letters on chemical graph theory in memory and honor of Professor Nenad Trinajstić