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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Reciprocal degree distance and graph properties
TL;DR: In this paper, sufficient conditions for a graph to be k -connected or β -deficient in terms of the reciprocal degree distance are given.
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Hamilton-connectedness and Hamilton-laceability of planar geometric graphs with applications
TL;DR: In this article, the authors used two different proof techniques to show the Hamilton-connectedness of graphs and constructed an infinite family of Hamilton connected convex polytope line graphs.
Journal ArticleDOI
Comparative results and bounds for the eccentric-adjacency index
Hongbo Hua,Kinkar Chandra Das +1 more
TL;DR: This paper establishes some sharp upper bounds on EAI for general connected graphs and quasi-trees, and investigates the relationship between AEDS and EAI, and proves that AEDs > EAi for any tree with at least three vertices.
Journal ArticleDOI
Bounding the $k$-Steiner Wiener and Wiener-Type Indices of Trees in Terms of Eccentric Sequence
TL;DR: In this article, it was shown that the unique trees that minimise the Wiener index among all trees with a given eccentric sequence were determined by the present authors, and they also showed that these results hold for a large class of distance-based topological indices which they termed Wiener-type indices.
Posted Content
Minimum Average Distance Triangulations
TL;DR: In this paper, the authors studied the problem of finding a triangulation T of a planar point set S such that the expected distance between two distinct points x and y chosen uniformly at random from S is minimized.
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.