标题： 关于随机图上拓扑和谱指数的分布 显示英文标题

标题： On the distribution of topological and spectral indices on random graphs

摘要： 我们针对在广泛的连通性状态下随机图$G=(V,E)$上的拓扑和谱指数分布进行了详细统计研究。 首先，我们考虑基于度数的拓扑指标（TI），并重点关注两类：$X_\Sigma(G) = \sum_{uv \in E} f(d_u,d_v)$和$X_\Pi(G) = \prod_{uv \in E} g(d_u,d_v)$，其中$uv$表示连接顶点$u$和$v$的$G$的边，$d_u$是顶点$u$的度数，并且$f(x,y)$和$g(x,y)$是顶点度数的函数。 具体来说，我们在Erdős-Rényi图和随机几何图上应用了$X_\Sigma(G)$和$X_\Pi(G)$，涵盖了从几乎孤立的顶点到主要连通图的整个转变过程。 我们验证了$P(X_\Sigma(G))$收敛于标准正态分布，同时证明了$P( X_\Pi(G))$收敛于对数正态分布。 此外，我们还分析了基于Revan度的指标和谱指标（这些指标由图邻接矩阵的特征值和特征向量定义）。 事实上，对于基于Revan度的指标，我们得到了与基于标准度的拓扑指标等效的结果。 相反，对于谱指标，我们报告了两种不同的模式：仅由特征值定义的指标的分布接近正态分布，而涉及特征值和特征向量的指标的分布则接近对数正态分布。 显示英文摘要

摘要： We perform a detailed statistical study of the distribution of topological and spectral indices on random graphs $G=(V,E)$ in a wide range of connectivity regimes. First, we consider degree-based topological indices (TIs), and focus on two classes of them: $X_\Sigma(G) = \sum_{uv \in E} f(d_u,d_v)$ and $X_\Pi(G) = \prod_{uv \in E} g(d_u,d_v)$, where $uv$ denotes the edge of $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $f(x,y)$ and $g(x,y)$ are functions of the vertex degrees. Specifically, we apply $X_\Sigma(G)$ and $X_\Pi(G)$ on Erd\"os-R\'enyi graphs and random geometric graphs along the full transition from almost isolated vertices to mostly connected graphs. While we verify that $P(X_\Sigma(G))$ converges to a standard normal distribution, we show that $P( X_\Pi(G))$ converges to a log-normal distribution. In addition we also analyze Revan-degree-based indices and spectral indices (those defined from the eigenvalues and eigenvectors of the graph adjacency matrix). Indeed, for Revan-degree indices, we obtain results equivalent to those for standard degree-based TIs. Instead, for spectral indices, we report two distinct patterns: the distribution of indices defined only from eigenvalues approaches a normal distribution, while the distribution of those indices involving both eigenvalues and eigenvectors approaches a log-normal distribution.