Wiener Index and Remoteness in Triangulations and Quadrangulations

É. Czabarka, P. Dankelmann, Trevor Olsen, L. Székely
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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.
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Wiener指数与三角形和四边形的距离
设$G$为连通图。连通图的维纳指数是所有无序顶点对之间距离的和。我们给出了具有给定连通性的简单三角剖分和四边形随着阶数增加的最大Wiener指数的渐近公式,并基于计算证据对三角剖分和四边形的极值进行了推测。如果$\overline{\sigma}(v)$表示$v$到$G$的所有其他顶点的距离的算术平均值,则将$G$的距离定义为$\overline{\sigma}(v)$对$G$的所有顶点$v$的最大值。我们给出了给定阶数和连通性的简单三角剖分和四边形的距离的明显上界。
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