Journal ArticleDOI
Wiener Index of Trees: Theory and Applications
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The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph as discussed by the authors, defined as the distance between all vertices in a graph.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.read more
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Journal ArticleDOI
Wiener Index of Hexagonal Systems
TL;DR: In this paper, the authors present the results known for W of the HS: method for computing W, expressions relating W with the structure of the respective HS, results on HS's extremal w.r.t. W, and on integers that cannot be the W-values of HS's.
Journal ArticleDOI
Resistance distance and the normalized Laplacian spectrum
Haiyan Chen,Fuji Zhang +1 more
TL;DR: Not only is it shown the resistance distance can be naturally expressed in terms of the normalized Laplacian eigenvalues and eigenvectors of G, but also a new index which is closely related to the spectrum of the Normalized LaPLacian is introduced.
Book ChapterDOI
Chemical Graph Theory
Ernesto Estrada,Danail Bonchev +1 more
TL;DR: This chapter on chemical graph theory forms part of the natural science and processes section of the handbook.
Journal ArticleDOI
The first and second Zagreb indices of some graph operations
TL;DR: Some exact expressions for the first and second Zagreb indices of graph operations containing the Cartesian product, composition, join, disjunction and symmetric difference of graphs will be presented.
A Survey on Graphs Extremal with Respect to Distance-Based Topological Indices
TL;DR: In this article, the authors present a survey on graphs extremal with respect to distance-based indices, with emphasis on the Wiener index, hyper-Wiener index and the Harary index.
References
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Journal ArticleDOI
Optimum Communication Spanning Trees
TL;DR: The cost of communication for a pair of nodes is multiplied by the sum of the distances of arcs which form the unique path connecting N_i and N_j in the spanning tree.
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On the sum of all distances in a graph or digraph
TL;DR: The transmission of a graph or digraph G is the sum of all distances in G and the independence of the transmission on the diameter or radius is shown.
Journal ArticleDOI
Extensions of the Wiener Number
TL;DR: Particularly for structure−property correlations there are many chemical graph-theoretic indices, one of which is Wiener's “path number”, which focuses on acyclic structure correlations.
Journal ArticleDOI
Mean distance in a graph
J. K. Doyle,Jack E. Graver +1 more
TL;DR: Borders for μ(Γ) are computed in terms of the number of vertices in Γ and the diameter of Γ to prove a formula for computing μ( Γ) whenΓ is a tree which is particularly useful when Γ has a high degree of symmetry.