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
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Further results on the expected hitting time, the cover cost and the related invariants of graphs
Jing Huang,Shuchao Li,Zheng Xie +2 more
TL;DR: In this paper, a close relation between hitting times of the simple random walk on a graph, the Kirchhoff index, the resistance centrality, and related invariants of unicyclic graphs is shown.
Journal ArticleDOI
Enumeration of BC-subtrees of trees
TL;DR: A BC-tree (block-cutpoint-tree) is a tree (with at least two vertices) where the distance between any two leaves is even.
Hyper-Wiener Index of Unicyclic Graphs
Rundan Xing,Bo Zhou,Xuli Qi +2 more
TL;DR: In this article, the authors determined the n-vertex unicyclic graphs of cycle length r with the smallest and the largest hyper-Wiener indices for 3 ≤ r ≤ n.
Journal ArticleDOI
Maximum values of Szeged index and edge-Szeged index of graphs☆
TL;DR: This note disprove two recent conjectures concerning with the maxi-mum value of Szeged index of graphs, which are due to Khalifeh et al. (Europ.
Journal ArticleDOI
On the Wiener index of generalized Fibonacci cubes and Lucas cubes
Sandi Klavzar,Yoomi Rho +1 more
TL;DR: It is proved that if for some d, the graph Q d ( f) is not isometric in Q d, then for any positive integer k, for almost all dimensions d ' the distance in Qd ' ( f ) can exceed the Hamming distance by k.
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.