{"title":"分割符号图的谱分析","authors":"Sandeep Kumar, Deepa Sinha","doi":"10.29020/nybg.ejpam.v17i1.4798","DOIUrl":null,"url":null,"abstract":"An ordered pair $\\Sigma = (\\Sigma^{u}$,$\\sigma$) is called the \\textit{signed graph}, where $\\Sigma^{u} = (V,E)$ is a \\textit{underlying graph} and $\\sigma$ is a signed mapping, called \\textit{signature}, from $E$ to the sign set $\\lbrace +, - \\rbrace$. The \\textit{splitting signed graph} $\\Gamma(\\Sigma)$ of a signed graph $\\Sigma$ is defined as, for every vertex $u \\in V(\\Sigma)$, take a new vertex $u'$. Join $u'$ to all the vertices of $\\Sigma$ adjacent to $u$ such that $\\sigma_{\\Gamma}(u'v) = \\sigma(u'v), \\ u \\in N(v)$. The objective of this paper is to propose an algorithm for the generation of a splitting signed graph, a splitting root signed graph from a given signed graph using Matlab. Additionally, we conduct a spectral analysis of the resulting graph. Spectral analysis is performed on the adjacency and laplacian matrices of the splitting signed graph to study its eigenvalues and eigenvectors. A relationship between the energy of the original signed graph $\\Sigma$ and the energy of the splitting signed graph $\\Gamma(\\Sigma)$ is established.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":"88 2","pages":""},"PeriodicalIF":16.4000,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Spectral Analysis of Splitting Signed Graph\",\"authors\":\"Sandeep Kumar, Deepa Sinha\",\"doi\":\"10.29020/nybg.ejpam.v17i1.4798\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"An ordered pair $\\\\Sigma = (\\\\Sigma^{u}$,$\\\\sigma$) is called the \\\\textit{signed graph}, where $\\\\Sigma^{u} = (V,E)$ is a \\\\textit{underlying graph} and $\\\\sigma$ is a signed mapping, called \\\\textit{signature}, from $E$ to the sign set $\\\\lbrace +, - \\\\rbrace$. The \\\\textit{splitting signed graph} $\\\\Gamma(\\\\Sigma)$ of a signed graph $\\\\Sigma$ is defined as, for every vertex $u \\\\in V(\\\\Sigma)$, take a new vertex $u'$. Join $u'$ to all the vertices of $\\\\Sigma$ adjacent to $u$ such that $\\\\sigma_{\\\\Gamma}(u'v) = \\\\sigma(u'v), \\\\ u \\\\in N(v)$. The objective of this paper is to propose an algorithm for the generation of a splitting signed graph, a splitting root signed graph from a given signed graph using Matlab. Additionally, we conduct a spectral analysis of the resulting graph. Spectral analysis is performed on the adjacency and laplacian matrices of the splitting signed graph to study its eigenvalues and eigenvectors. A relationship between the energy of the original signed graph $\\\\Sigma$ and the energy of the splitting signed graph $\\\\Gamma(\\\\Sigma)$ is established.\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":\"88 2\",\"pages\":\"\"},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-01-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.29020/nybg.ejpam.v17i1.4798\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.29020/nybg.ejpam.v17i1.4798","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
An ordered pair $\Sigma = (\Sigma^{u}$,$\sigma$) is called the \textit{signed graph}, where $\Sigma^{u} = (V,E)$ is a \textit{underlying graph} and $\sigma$ is a signed mapping, called \textit{signature}, from $E$ to the sign set $\lbrace +, - \rbrace$. The \textit{splitting signed graph} $\Gamma(\Sigma)$ of a signed graph $\Sigma$ is defined as, for every vertex $u \in V(\Sigma)$, take a new vertex $u'$. Join $u'$ to all the vertices of $\Sigma$ adjacent to $u$ such that $\sigma_{\Gamma}(u'v) = \sigma(u'v), \ u \in N(v)$. The objective of this paper is to propose an algorithm for the generation of a splitting signed graph, a splitting root signed graph from a given signed graph using Matlab. Additionally, we conduct a spectral analysis of the resulting graph. Spectral analysis is performed on the adjacency and laplacian matrices of the splitting signed graph to study its eigenvalues and eigenvectors. A relationship between the energy of the original signed graph $\Sigma$ and the energy of the splitting signed graph $\Gamma(\Sigma)$ is established.
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.