Abstract
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.
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Dobrynin, A.A., Entringer, R. & Gutman, I. Wiener Index of Trees: Theory and Applications. Acta Applicandae Mathematicae 66, 211–249 (2001). https://doi.org/10.1023/A:1010767517079
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DOI: https://doi.org/10.1023/A:1010767517079