论文标题
Erdös-Rényi模型中Randić索引的计算和分析研究
Computational and analytical studies of the Randić index in Erdös-Rényi models
论文作者
论文摘要
在这项工作中，我们对Erdös-rényi型号$ g（n，p）$ r（g）$ r（g）$ r（n，p）进行计算和分析研究，其特征在于$ n $ vertices与概率$ p \ in（0,1）$独立连接。首先，从详细的缩放分析中，我们表明$ \ weft \ langle \ edline {r}（g）\ right \ rangle = \ weled = \ left \ langle r（g）\ rangle/（n/2）$用产品$之一$ np $定义$ np np np np np np np npecime np np np np np np npecimes： （$ r（g）\大约0 $），$ 0.01 <ξ<10 $（其中$ 0 <r（g）<n/2 $）的过渡制度，以及几乎完全完整的图表的$ξ> 10 $（$ r（g）\ of（g）\大约n/2 $）的制度。然后，由$ \ left \ langle \ coperline {r}（g）\ right \ rangle $的缩放的动机，我们通过分析（i）获得将$ r（g）$连接到其他拓扑指数的新关系（i）与获得这些结果相对于获得这些结果的$ R（ii）的新关系，这些图形是为了获得这些结果，以获取$ n的模型（ii）$ n $ n $ n（g）$ r n（g）$ r n yex $ r n yex in grand ynex in（g）。
In this work we perform computational and analytical studies of the Randić index $R(G)$ in Erdös-Rényi models $G(n,p)$ characterized by $n$ vertices connected independently with probability $p \in (0,1)$. First, from a detailed scaling analysis, we show that $\left\langle \overline{R}(G) \right\rangle = \left\langle R(G)\right\rangle/(n/2)$ scales with the product $ξ\approx np$, so we can define three regimes: a regime of mostly isolated vertices when $ξ< 0.01$ ($R(G)\approx 0$), a transition regime for $0.01 < ξ< 10$ (where $0<R(G)< n/2$), and a regime of almost complete graphs for $ξ> 10$ ($R(G)\approx n/2$). Then, motivated by the scaling of $\left\langle \overline{R}(G) \right\rangle$, we analytically (i) obtain new relations connecting $R(G)$ with other topological indices and characterize graphs which are extremal with respect to the relations obtained and (ii) apply these results in order to obtain inequalities on $R(G)$ for graphs in Erdös-Rényi models.
