Journal ArticleDOI
Wiener Index of Trees: Theory and Applications
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TLDR
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
Citations
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Journal ArticleDOI
Comparison between the Wiener index and the Zagreb indices and the eccentric connectivity index for trees
TL;DR: The Wiener index and the Zagreb indices and the eccentric connectivity index for trees are compared to find the most suitable indices for graphs and trees.
Maximum Wiener Index of Trees With Given Segment Sequence
TL;DR: In this article, the authors considered the problem of maximizing the Wiener index among trees with given segment sequence or number of segments, and they showed that the maximum is always obtained for a so-called quasi-caterpillar, and further characterized its structure.
Journal ArticleDOI
The Wiener index of the kth power of a graph
Xinhui An,Baoyindureng Wu +1 more
TL;DR: The Nordhaus–Gaddum-type inequality for the Wiener index of the graph G k is presented and the bounds on theWiener index are given.
Journal ArticleDOI
The distances between internal vertices and leaves of a tree
TL;DR: A relatively comprehensive study of distance-based graph invariants, identifying the ''middle part'' of a tree with respect to the total distance from leaves and providing extremal trees under different constraints that maximize or minimize the sum of all such distances.
Journal ArticleDOI
On a relation between Szeged and Wiener indices of bipartite graphs
TL;DR: In this paper, Hansen et al. showed that the Szeged index and Wiener index of a connected bipartite graph, with vertices and edges, can be computed in 4n-8 time.
References
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Journal ArticleDOI
Laplacian matrices of graphs: a survey
TL;DR: In this paper, the authors survey some of the many results known for Laplacian matrices, and present a survey of the most important results in the field of graph analysis.
THE LAPLACIAN SPECTRUM OF GRAPHS y
TL;DR: A survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Lapla-cian eigenvalue 2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidth-type parameters of a graph is given in this article.
Book
Mathematical concepts in organic chemistry
Ivan Gutman,Oskar E. Polansky +1 more
TL;DR: In this paper, the authors define the topology of a graph as follows: 1.1 Topology in Chemistry, 2.2 Geometry, Symmetry, Topology, Graph Automorphisms, and Graph Topology.