5 research outputs found

    Wiener Index and Remoteness in Triangulations and Quadrangulations

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    Let GG 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 σ(v)\overline{\sigma}(v) denotes the arithmetic mean of the distances from vv to all other vertices of GG, then the remoteness of GG is defined as the largest value of σ(v)\overline{\sigma}(v) over all vertices vv of GG. We give sharp upper bounds on the remoteness of simple triangulations and quadrangulations of given order and connectivity

    Proximity, remoteness and maximum degree in graphs

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    The average distance of a vertex vv of a connected graph GG is the arithmetic mean of the distances from vv to all other vertices of GG. The proximity π(G)\pi(G) and the remoteness ρ(G)\rho(G) of GG are the minimum and the maximum of the average distances of the vertices of GG, respectively. In this paper, we give upper bounds on the remoteness and proximity for graphs of given order, minimum degree and maximum degree. Our bounds are sharp apart from an additive constant.Comment: 20 page

    Proximity and Remoteness in Graphs: a survey

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    The proximity π=π(G)\pi = \pi (G) of a connected graph GG is the minimum, over all vertices, of the average distance from a vertex to all others. Similarly, the maximum is called the remoteness and denoted by ρ=ρ(G)\rho = \rho (G). The concepts of proximity and remoteness, first defined in 2006, attracted the attention of several researchers in Graph Theory. Their investigation led to a considerable number of publications. In this paper, we present a survey of the research work.Comment: arXiv admin note: substantial text overlap with arXiv:1204.1184 by other author
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