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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On the Laplacian coefficients of unicyclic graphs with prescribed matching number
TL;DR: The unique connected unicyclic graph with the minimal Laplacian coefficients among all connected unicycleclic graphs of order n except S'n, where S"n^' is the unicyCLic graph obtained from the n-vertex star S" n by joining two of its pendent vertices with an edge.
Journal ArticleDOI
On the Laplacian coefficients of trees with a perfect matching
Shang-wang Tan,Tian-mei Song +1 more
TL;DR: In this article, a graphic transformation that increases all Laplacian coefficients of an arbitrary tree is given, and the minimum element and the second minimum element in Γ(n) under the partial order ⪯, and finally the maximum element and second maximum element in the set of all trees with 2n vertices.
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Digraphs with large maximum Wiener index
TL;DR: It is shown that among digraphs on n vertices, the directed cycle C ?
Journal ArticleDOI
Distance‐based topological descriptors for measuring the π‐electronic energy of benzenoid hydrocarbons with applications to carbon nanotubes
Journal ArticleDOI
A few families of Cayley graphs and their efficiency as communication networks
TL;DR: An algorithm for computing shortest paths and the exact value of their diameters is given for the family of circulant graphs of degree 4 and lower and upper bounds for their forwarding and optical indices are obtained.
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.