Wiener Index and Remoteness in Triangulations and Quadrangulations
Authors: Éva Czabarka ; Peter Dankelmann ; Trevor Olsen ; László A. Székely
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Éva Czabarka;Peter Dankelmann;Trevor Olsen;László A. Székely
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.