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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: SPα(G) = ∑uv∈E(G) du, if du = dv, and SPα(G) = ∑uv∈E(G)[(dαu − dαv ) / (α(du − dv)]1/(α−1), otherwise. Here, uv denotes the edge of the network G connecting the vertices u and v, du is the degree of the vertex u, and α ∈ ℝ\{0, 1}. We also consider the limiting cases SPα→0(G) and SPα→1(G) that we name as the logarithmic–mean index and the identric–mean index, respectively. Indeed, for given values of α, 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 〈SPα(G)〉, normalized to the order of the network, scale with the corresponding average degree 〈d〉. Some mathematical properties of the Stolarsky–Puebla index are also discussed.