Stolarsky-Puebla指数
Stolarsky-Puebla index
摘要：
我们引入了一个基于度的可变拓扑指数的启发Stolarsky平均（称为推广的对数平均）。我们将这个新索引命名为Stolarsky-Puebla索引：如果是$d_u=d_v$，则命名为$SP_α(G) = \sum_{uv \in E(G)} d_u$，否则命名为$SP_α(G) = \sum_{uv \in E(G)} \left[\left( d_u^α-d_v^α\right)/\left( α(d_u-d_v\right)\right]^{1/(α-1)}$。这里，$uv$表示连接顶点$u$和$v$的网络$G$的边，$d_u$是顶点$u$的度，而$α\in \mathbb{R} \backslash \{0,1\}$。事实上，对于给定的$α$值，Stolarsky-Puebla指数再现了著名的拓扑指数，如倒数Randic指数，第一个Zagreb指数和几个平均Sombor指数。此外，我们将这些指标应用于随机网络，并证明了$\left< SP_α(G) \right>$，归一化到网络的顺序，规模与相应的平均度$\left< d \right>$。
英文摘要：
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: $SP_α(G) = \sum_{uv \in E(G)} d_u$, if $d_u=d_v$, and $SP_α(G) = \sum_{uv \in E(G)} \left[\left( d_u^α-d_v^α\right)/\left( α(d_u-d_v\right)\right]^{1/(α-1)}$, 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 $α\in \mathbb{R} \backslash \{0,1\}$. 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 $\left< SP_α(G) \right>$, normalized to the order of the network, scale with the corresponding average degree $\left< d \right>$.
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