9 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

    Inverse Problems Related to the Wiener and Steiner-Wiener Indices

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    In a graph, the generalized distance between multiple vertices is the minimum number of edges in a connected subgraph that contains these vertices. When we consider such distances between all subsets of kk vertices and take the sum, it is called the Steiner kk-Wiener index and has important applications in Chemical Graph Theory. In this thesis we consider the inverse problems related to the Steiner Wiener index, i.e. for what positive integers is there a graph with Steiner Wiener index of that value

    The Maximum Wiener Index of Maximal Planar Graphs

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    The Wiener index of a connected graph is the sum of the distances between all pairs of vertices in the graph. It was conjectured that the Wiener index of an nn-vertex maximal planar graph is at most ⌊118(n3+3n2)⌋\lfloor\frac{1}{18}(n^3+3n^2)\rfloor. We prove this conjecture and for every nn, n≥10n \geq 10, determine the unique nn-vertex maximal planar graph for which this maximum is attained.Comment: 13 pages, 4 figure

    Extremal Values of Ratios: Distance Problems vs. Subtree Problems in Trees

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    The authors discovered a dual behaviour of two tree indices, the Wiener index and the number of subtrees, for a number of extremal problems [Discrete Appl. Math. 155 (3) 2006, 374-385; Adv. Appl. Math. 34 (2005), 138-155]. Barefoot, Entringer and Székely [Discrete Appl. Math. 80 (1997), 37-56] determined extremal values of σT(w)/σT(u), σT(w)/σT(v), σ(T)/σT(v), and σ(T)/σT(w), where T is a tree on n vertices, v is in the centroid of the tree T, and u,w are leaves in T. In this paper we test how far the negative correlation between distances and subtrees go if we look for the extremal values of FT(w)/FT(u), FT(w)/FT(v), F(T)/FT(v), and F(T)/FT(w), where T is a tree on n vertices, v is in the subtree core of the tree T, and u,w are leaves in T-the complete analogue of [Discrete Appl. Math. 80 (1997), 37-56], changing distances to the number of subtrees. We include a number of open problems, shifting the interest towards the number of subtrees in graphs

    Molecular graphs and the inverse Wiener index problem

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