Journal ArticleDOI
Wiener Index of Trees: Theory and Applications
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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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On the transmission of uniform unicyclic hypergraphs
Hongying Lin,Bo Zhou +1 more
TL;DR: In this paper, the transmission of a connected hypergraph is defined as the summation of distances between all unordered pairs of distinct vertices, and the unique uniform unicyclic hypergraphs of fixed size with minimum and maximum transmissions, respectively, are determined.
Journal ArticleDOI
Comparing Eccentricity-Based Graph Invariants
TL;DR: It is proved that EDS ≥ EM1 for any connected graph, whereas EDS > EM2 for trees, and in the case of trees, EM1 ≥ CEI, whereas EM2 > CEI for trees with at least three vertices.
Bounds on differences between some graph theoretic invariants
TL;DR: The graph invariants involved in the present work are the proximity, the remoteness, the eccentricity, the average distance, the frequencies of the maximum and minimum degrees, the domination number, the stability number and the chromatic number as discussed by the authors.
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The connective eccentricity index and modified second Zagreb index of Parikh word representable graphs
Hongbo Hua,Maolin Wang +1 more
TL;DR: In this article, the authors present explicit formulas for the connective eccentricity index and modified second Zagreb index of the Parikh word representable graphs corresponding to binary core words of the form aub over a binary alphabet.
Journal ArticleDOI
Comparative study of distance-based graph invariants
TL;DR: In this paper, Liberti et al. investigated the relationship between RDD and other three graph invariants AEDS, CEI and AD, and showed that AEDs>RDD for any tree with at least three vertices.
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.