arXiv 2210.047492022-10-10math.CO

随机图的Revan度指标

Revan-degree indices on random graphs

R. Aguilar-Sanchez, I. F. Herrera-Gonzalez, J. A. Mendez-Bermudez, Jose M. Sigarreta

备注：: 16 pages, 10 figures

摘要：

给定一个简单连通无向图$G=(V(G),E(G))$，我们考虑两类图不变量：$RX_Σ(G) = \sum_{uv \in E(G)} F(r_u,r_v)$（最近引起了人们的兴趣）和$RX_Π(G) = \prod_{uv \in E(G)} F(r_u,r_v)$（我们在本工作中引入的）;其中$uv$表示连接顶点$u$和$v$的$G$的边，$r_u$是顶点$u$的Revan度，$F$是Revan顶点度的函数.这里，具有$Δ$和$δ$的$r_u = Δ+ δ- d_u$是$G$和$d_u$的顶点之间的最大和最小度是顶点$u$的度。特别地，我们将$RX_Σ(G)$和R $X_Π(G)$应用于两种随机图模型：Erdös-Rényi图和随机几何图。通过一个彻底的计算研究，我们表明，$\left< RX_Σ(G) \right>$和$\left< \ln RX_Π(G) \right>$，规范化的顺序的图形，规模与平均瑞文度$\left< r \right>$ ;这里$\left< \cdot \right>$表示的平均超过一个合奏的随机图。此外，我们提供了几个图不变量的密集图极限的两个家庭的解析表达式。

英文摘要：

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.

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英文摘要：

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