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
Citations
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Upgrading the Wiener index
TL;DR: The Wiener index W is the oldest molecular graph-based structure-descriptor as discussed by the authors, defined as the sum of the distances of all pairs of vertices of the molecular graph G, where the distance is the number of edges in the shortest path connecting the respective vertices, and where G is the hydrogen-depleted molecular graph.
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Complete solution of equation for the Wiener index of iterated line graphs of trees
TL;DR: It is shown that there is an infinite class T of trees T satisfying W ( L 3 ( T) ) = W ( T ) , which disproves a conjecture of Dobrynin and Entringer.
The Schultz Molecular Topological Index of Polyhex Nanotubes 1
TL;DR: In this paper, the Schultz molecular topological index in polyhex nanotubes is calculated and a formula for calculating the Schultz topological indices is given for calculating polyhex topology.
Journal ArticleDOI
Sharp Bounds and Normalization of Wiener-Type Indices
Dechao Tian,Kwok Pui Choi +1 more
TL;DR: The normalized -Wiener indices were demonstrated to improve significantly the hierarchical clustering over the non-normalized counterparts, and the maximum and minimum of over a set of networks with nodes are identified.
Journal ArticleDOI
On extremal cacti with respect to the edge Szeged index and edge-vertex Szeged index
TL;DR: In this paper, the P edge Szeged index and the edge-vertex index for cacti with order n and k======consuming cycles were determined, and all the graphs that achieved the lower bounds were identified.
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.