É. Czabarka, P. Dankelmann, Trevor Olsen, L. Székely
{"title":"Wiener Index and Remoteness in Triangulations and Quadrangulations","authors":"É. Czabarka, P. Dankelmann, Trevor Olsen, L. Székely","doi":"10.46298/dmtcs.6473","DOIUrl":null,"url":null,"abstract":"Let $G$ be a a connected graph. The Wiener index of a connected graph is the\nsum of the distances between all unordered pairs of vertices. We provide\nasymptotic formulae for the maximum Wiener index of simple triangulations and\nquadrangulations with given connectivity, as the order increases, and make\nconjectures for the extremal triangulations and quadrangulations based on\ncomputational evidence. If $\\overline{\\sigma}(v)$ denotes the arithmetic mean\nof the distances from $v$ to all other vertices of $G$, then the remoteness of\n$G$ is defined as the largest value of $\\overline{\\sigma}(v)$ over all vertices\n$v$ of $G$. We give sharp upper bounds on the remoteness of simple\ntriangulations and quadrangulations of given order and connectivity.\n","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"42 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"12","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discret. Math. Theor. Comput. Sci.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.46298/dmtcs.6473","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 12
Abstract
Let $G$ be a a connected graph. The Wiener index of a connected graph is the
sum of the distances between all unordered pairs of vertices. We provide
asymptotic formulae for the maximum Wiener index of simple triangulations and
quadrangulations with given connectivity, as the order increases, and make
conjectures for the extremal triangulations and quadrangulations based on
computational evidence. If $\overline{\sigma}(v)$ denotes the arithmetic mean
of the distances from $v$ to all other vertices of $G$, then the remoteness of
$G$ is defined as the largest value of $\overline{\sigma}(v)$ over all vertices
$v$ of $G$. We give sharp upper bounds on the remoteness of simple
triangulations and quadrangulations of given order and connectivity.