{"title":"基于度的拓扑指数的渐近分布","authors":"Mingao Yuan","doi":"10.46793/match.91-1.135y","DOIUrl":null,"url":null,"abstract":"Topological indices play a significant role in mathematical chemistry. Given a graph $\\mathcal{G}$ with vertex set $\\mathcal{V}=\\{1,2,\\dots,n\\}$ and edge set $\\mathcal{E}$, let $d_i$ be the degree of node $i$. The degree-based topological index is defined as $\\mathcal{I}_n=$ $\\sum_{\\{i,j\\}\\in \\mathcal{E}}f(d_i,d_j)$, where $f(x,y)$ is a symmetric function. In this paper, we investigate the asymptotic distribution of the degree-based topological indices of a heterogeneous Erd\\H{o}s-R\\'{e}nyi random graph. We show that after suitably centered and scaled, the topological indices converges in distribution to the standard normal distribution. Interestingly, we find that the general Randi\\'{c} index with $f(x,y)=(xy)^{\\tau}$ for a constant $\\tau$ exhibits a phase change at $\\tau=-\\frac{1}{2}$.","PeriodicalId":51115,"journal":{"name":"Match-Communications in Mathematical and in Computer Chemistry","volume":"125 1","pages":"0"},"PeriodicalIF":2.9000,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Asymptotic Distribution of Degree-Based Topological Indices\",\"authors\":\"Mingao Yuan\",\"doi\":\"10.46793/match.91-1.135y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Topological indices play a significant role in mathematical chemistry. Given a graph $\\\\mathcal{G}$ with vertex set $\\\\mathcal{V}=\\\\{1,2,\\\\dots,n\\\\}$ and edge set $\\\\mathcal{E}$, let $d_i$ be the degree of node $i$. The degree-based topological index is defined as $\\\\mathcal{I}_n=$ $\\\\sum_{\\\\{i,j\\\\}\\\\in \\\\mathcal{E}}f(d_i,d_j)$, where $f(x,y)$ is a symmetric function. In this paper, we investigate the asymptotic distribution of the degree-based topological indices of a heterogeneous Erd\\\\H{o}s-R\\\\'{e}nyi random graph. We show that after suitably centered and scaled, the topological indices converges in distribution to the standard normal distribution. Interestingly, we find that the general Randi\\\\'{c} index with $f(x,y)=(xy)^{\\\\tau}$ for a constant $\\\\tau$ exhibits a phase change at $\\\\tau=-\\\\frac{1}{2}$.\",\"PeriodicalId\":51115,\"journal\":{\"name\":\"Match-Communications in Mathematical and in Computer Chemistry\",\"volume\":\"125 1\",\"pages\":\"0\"},\"PeriodicalIF\":2.9000,\"publicationDate\":\"2023-10-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Match-Communications in Mathematical and in Computer Chemistry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.46793/match.91-1.135y\",\"RegionNum\":2,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Match-Communications in Mathematical and in Computer Chemistry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.46793/match.91-1.135y","RegionNum":2,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
Asymptotic Distribution of Degree-Based Topological Indices
Topological indices play a significant role in mathematical chemistry. Given a graph $\mathcal{G}$ with vertex set $\mathcal{V}=\{1,2,\dots,n\}$ and edge set $\mathcal{E}$, let $d_i$ be the degree of node $i$. The degree-based topological index is defined as $\mathcal{I}_n=$ $\sum_{\{i,j\}\in \mathcal{E}}f(d_i,d_j)$, where $f(x,y)$ is a symmetric function. In this paper, we investigate the asymptotic distribution of the degree-based topological indices of a heterogeneous Erd\H{o}s-R\'{e}nyi random graph. We show that after suitably centered and scaled, the topological indices converges in distribution to the standard normal distribution. Interestingly, we find that the general Randi\'{c} index with $f(x,y)=(xy)^{\tau}$ for a constant $\tau$ exhibits a phase change at $\tau=-\frac{1}{2}$.
期刊介绍:
MATCH Communications in Mathematical and in Computer Chemistry publishes papers of original research as well as reviews on chemically important mathematical results and non-routine applications of mathematical techniques to chemical problems. A paper acceptable for publication must contain non-trivial mathematics or communicate non-routine computer-based procedures AND have a clear connection to chemistry. Papers are published without any processing or publication charge.