论文标题
随机图上的Revan-Degree索引
Revan-degree indices on random graphs
论文作者
论文摘要
给定一个简单连接的非导向图$ g =（v（g），e（g））$，我们考虑两个图形不变的家族：$rx_σ（g）= \ sum_ {uv \ in E（g）} f（r_u，r_v）$ f（r_u，r_v）$（我们在这项工作中介绍）;其中$ uv $表示连接顶点$ u $和$ v $的$ g $的边缘，$ r_u $是顶点$ u $的revan度，而$ f $是Revan Vertex学位的函数。在这里，$ r_u =δ+δ-d_u $带有$δ$，$δ$和$ g $和$ d_u $的最高和最小度是顶点$ u $的程度。特别是，我们在两个随机图的型号上同时应用$rx_σ（g）$和r $x_π（g）$：erdös-rényi图和随机几何图。通过一项彻底的计算研究，我们表明$ \ weft <rx_σ（g）\ right> $和$ \ weft <\ lnrx_π（g）\ right> $，标准化为图表的顺序，平均revan级$ \ left $ \ left <r \ right> $;这里$ \ left <\ cdot \ right> $表示随机图的合奏上的平均值。此外，我们在密集的图限制下为两个家族的几个图形不变性提供了分析表达式。
Given a simple connected non-directed graph $G=(V(G),E(G))$, we consider two families of graph invariants: $RX_Σ(G) = \sum_{uv \in E(G)} F(r_u,r_v)$ (which has gained interest recently) and $RX_Π(G) = \prod_{uv \in E(G)} F(r_u,r_v)$ (that we introduce in this work); where $uv$ denotes the edge of $G$ connecting the vertices $u$ and $v$, $r_u$ is the Revan degree of the vertex $u$, and $F$ is a function of the Revan vertex degrees. Here, $r_u = Δ+ δ- d_u$ with $Δ$ and $δ$ the maximum and minimum degrees among the vertices of $G$ and $d_u$ is the degree of the vertex $u$. Particularly, we apply both $RX_Σ(G)$ and R$X_Π(G)$ on two models of random graphs: Erdös-Rényi graphs and random geometric graphs. By a thorough computational study we show that $\left< RX_Σ(G) \right>$ and $\left< \ln RX_Π(G) \right>$, normalized to the order of the graph, scale with the average Revan degree $\left< r \right>$; here $\left< \cdot \right>$ denotes the average over an ensemble of random graphs. Moreover, we provide analytical expressions for several graph invariants of both families in the dense graph limit.
