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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On the extremal properties of the average eccentricity
TL;DR: In this article, the authors analyze extremal properties of the average eccentricity, and introduce two graph transformations that increase or decrease the eccentricity of a given graph, and resolve four conjectures obtained by the system AutoGraphiX.
Journal ArticleDOI
Variable neighborhood search for extremal graphs. 10. Comparison of irregularity indices for chemical trees.
TL;DR: In this paper, the authors studied the properties of chemical trees and derived analytical expressions for extremal values and extremal graphs for three indices: Collatz-Sinogowitz, of Albertson, and of Bell's variance of degrees.
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Correlation of Graph-Theoretical Indices
TL;DR: It is shown how an asymptotic formula for the correlation coefficient can be obtained when simply generated families of trees are investigated and for rooted ordered trees, the calculations are done explicitly.
Journal ArticleDOI
Distance spectral radius of trees with given matching number
TL;DR: A conjecture for the maximal case based on the computer search among trees on n@?24 vertices is posed and the extremal tree that uniquely maximizes the distance spectral radius among n-vertex trees with perfect matching and fixed maximum degree @D is found.
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.