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 $, normalized to the order of the network, scale with the corresponding average degree $\left $.
Highlights
For two positive real numbers x, y, the Stolarsky mean Sα(x, y) is defined as [22]x if x = y, ξα − ηα 1/(α−1)Sα(x, y) = (ξ,η)l→ im(x,y) α(ξ − η) = xα − yα 1/(α−1) otherwise, (1)α(x − y) here, α ∈ R\{0, 1}
F are currently being studied in mathematical chemistry; where uv denotes the edge of the graph G connecting the vertices u and v, du is the degree of the vertex u, and F (x, y) is an appropriate chosen function, see e.g. [12]
Α(du − dv) where uv denotes the edge of the graph G connecting the vertices u and v, du is the degree of the vertex u, and α ∈ R\{0, 1}
Summary
For two positive real numbers x, y, the Stolarsky mean Sα(x, y) is defined as [22]. α(x − y) here, α ∈ R\{0, 1}. For two positive real numbers x, y, the Stolarsky mean Sα(x, y) is defined as [22]. Sα(x, y) is known as the generalization of the logarithmic mean [16]. For given values of α, Sα(x, y) reproduces known means including the logarithmic mean, when α → 0, and some cases of the power mean [5, 23]. There is a well-known inequality relating the Stolarsky mean and the power mean, namely [6, 16, 19]: S−1(x, y) = P Mα→0(x, y) ≤ Sα→0(x, y) ≤ P M1/3(x, y) ≤ S2(x, y) = P M1(x, y). −∞ Sα→−∞(x, y) = min(x, y) minimum value, P Mα→−∞(x, y) x3 + x2y + xy2 + y3 −1/5. 4 S4(x, y) = 4 ∞ Sα→∞(x, y) = max(x, y) arithmetic mean, P M1(x, y) maximum value, P Mα→∞(x, y)
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