![One graph gives two planar maps A tree is a connected graph without cycle. A plan tree is a tree designed on the plane [4]. The Wiener index of a connected graph is the sum of](https://www.researchgate.net/profile/Mohamed-El-Marraki-2/publication/261552527/figure/fig1/AS:296701198913549@1447750487411/One-graph-gives-two-planar-maps-A-tree-is-a-connected-graph-without-cycle-A-plan-tree-is_Q320.jpg)
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Journal of Theoretical and Applied Information Technology WIENER INDEX OF PLANAR MAPS
Abstract and Figures
In trees with n vertices, the Wiener index of tree is minimized by stars and maximized by paths, both uniquely. In this paper, we give an inequality similar in the case of planar maps.
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A method for the calculation of the Wiener number (W) of trees is outlined, by means of which it is relatively easy to find general formulas for W of various classes of branched molecules. The method is applied to dendrimers, whose molecular graphs are highly branched trees.
The Wiener index is the sum of topological distances between all pairs of vertices in a connected graph such as represents the structural formula of a molecule. Firstly, we investigate some properties of the partially ordered set of all vectors associated with a tree with respect to majorization. Then these results are used to characterize the trees which minimize the Wiener index among all trees with a given degree sequence. Consequently, all extremal trees with the smallest Wiener index are obtained in the sets of all trees of order n with the maximum degree, the leaf number and the matching number respectively.
The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph. The paper outlines the results known for W of trees: methods for computation of W and combinatorial expressions for W for various classes of trees, the isomorphism–discriminating power of W, connections between W and the center and centroid of a tree, as well as between W and the Laplacian eigenvalues, results on the Wiener indices of the line graphs of trees, on trees extremal w.r.t. W, and on integers which cannot be Wiener indices of trees. A few conjectures and open problems are mentioned, as well as the applications of W in chemistry, communication theory and elsewhere.
Asymptotics are obtained for the mean, variance and higher moments as well as for the distribution of the Wiener index of a random tree from a simply generated family (or, equivalently, a critical Galton-- Watson tree). We also establish a joint asymptotic distribution of the Wiener index and the internal path length, as well as asymptotics for the covariance and other mixed moments. The limit laws are described using functionals of a Brownian excursion. The methods include both Aldous' theory of the continuum random tree and analysis of generating functions. 1.
The distance between two vertices of a tree
- M El Marraki
- A Modabish
- G Al Hagri
M. El Marraki, A. Modabish and G. Al Hagri,
"The distance between two vertices of a tree",
to be published.