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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Optimising Robustness of Consensus to Noise on Directed Networks
TL;DR: This dissertation focuses on understanding the role played by a directed communication graph in the ability of a group to maintain consensus in noisy environments, and derives rules by which local changes can be made that will guarantee that the robustness of the entire system will improve.
Proceedings ArticleDOI
Distances and isomorphisms in 4-regular circulant graphs
Alfredo Donno,Donatella Iacono +1 more
TL;DR: In this article, the Hosoya polynomial of the Cayley graph of cyclic groups is computed and the order 17 is the smallest case providing two non-isomorphic 4-regular circulant graphs with the same Wiener index.
Journal Article
The Laplacian Spectra of Graphs and Complex Networks
TL;DR: The paper is a brief survey of some recent new results and progress of the Laplacian spectra of graphs and complex networks (in particular, random graph and the small world network).
Journal ArticleDOI
Wiener Index of Graphs and Their Line Graphs
Xiaohai Su,Ligong Wang,Yun Gao +2 more
TL;DR: In this paper, it was shown that for λ = 2 there is an infinite family of planar bipartite chemical graphs G of girth 4 with the cyclomatic number λ, but their line graphs are not chemical graphs.
The Schultz Molecular Topological Index of C4C8 Nanotubes 1
Lixin Xu,China Hanyuan Deng +1 more
TL;DR: In this article, a method for calculating the Schultz molecular topological index (MTI) of C4C8 nanotubes has been presented, which can cover either a cylinder or a torus.
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.