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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The expected values of Wiener indices in random polycyclic chains
Hanlin Chen,Chao-Jun Li +1 more
TL;DR: In this article , the expected value of the Wiener index in random polycyclic chains with n polygons has been analyzed, where n is the number of vertices in the graph.
Journal ArticleDOI
Investments in social ties, risk sharing, and inequality
Attila Ambrus,Matthew Elliott +1 more
TL;DR: In this paper, the authors investigate stable and efficient networks in the context of risk sharing, when it is costly to establish and maintain relationships that facilitate risk sharing and find that the most stable efficient networks also generate the most inequality.
Journal ArticleDOI
The maximum Wiener index of maximal planar graphs
Debarun Ghosh,Ervin Győri,Ervin Győri,Addisu Paulos,Addisu Paulos,Nika Salia,Nika Salia,Oscar Zamora,Oscar Zamora +8 more
TL;DR: This paper determines the unique n -vertex maximal planar graph attaining this maximum, for every n ≥ 10, and proves the Wiener index is the sum of the distances between all pairs of vertices.
Journal Article
Hosoya polynomial of Hanoi graphs
TL;DR: A recursive method for the calculation of the Hosoya polynomial of Hanoi graph is designed, making it possible to compute various distance{based invariants of H Vietnam graphs.
Posted Content
Properties of the Hyper-Wiener index as a local function
TL;DR: In this article, the authors considered the local version of the Hyper-Wiener index, defined as $ww_G(v)=\sum\limits_{u\in V(G)}(d^2(u,v)+d(u-v))$ for a vertex $v$ in a graph $G$, in trees.
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.