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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Relation Between Randic Index and Average Distance of Trees 1
Marek Cygan,Michał Pilipczuk +1 more
TL;DR: In this article, it was shown that for any tree T with n 1 leaves R(T ) ≥ ad(T +m ax(0, √ n 1 − 2), where ad is the average distance between vertices of T. As a consequence, the conjecture R(G) ≥ ad (G) given by Fajtlowicz in 1988 for the case when G is a tree was resolved.
Journal ArticleDOI
A lower bound of revised Szeged index of bicyclic graphs
TL;DR: The lower bound of revised Szeged index among all bicyclic graphs is identified, and the extremal graphs that attain the lower bound are characterized.
Journal ArticleDOI
Extremal hitting times of trees with some given parameters
Huihui Zhang,Shuchao Li +1 more
TL;DR: In this paper, the hitting time HG(x,y) is defined as the expected number of steps that a simple random walk takes to go from x to y in a connected graph G=(VG,EG) with x,y∈VG.
Posted Content
Graphs with the second and third maximum Wiener index over the 2-vertex connected graphs
TL;DR: It is shown that the second maximum graph is obtained from C_n by introducing a new edge that connects two vertices at distance two on the cycle if $n
e 6$ if the cycle attains the maximum value of Wiener index.
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.