Wiener Index of Trees: Theory and Applications
TL;DR: 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.
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TL;DR: In this paper, the authors present the results known for W of the HS: method for computing W, expressions relating W with the structure of the respective HS, results on HS's extremal w.r.t. W, and on integers that cannot be the W-values of HS's.
Abstract: The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph. Hexagonal systems (HS's) are a special type of plane graphs in which all faces are bounded by hexagons. These provide a graph representation of benzenoid hydrocarbons and thus find applications in chemistry. The paper outlines the results known for W of the HS: method for computation of W, expressions relating W with the structure of the respective HS, results on HS's extremal w.r.t. W, and on integers that cannot be the W-values of HS's. A few open problems are mentioned. The chemical applications of the results presented are explained in detail.
371 citations
TL;DR: Not only is it shown the resistance distance can be naturally expressed in terms of the normalized Laplacian eigenvalues and eigenvectors of G, but also a new index which is closely related to the spectrum of the Normalized LaPLacian is introduced.
294 citations
11 May 2018
TL;DR: This chapter on chemical graph theory forms part of the natural science and processes section of the handbook.
Abstract: This chapter on chemical graph theory forms part of the natural science and processes section of the handbook
263 citations
TL;DR: Some exact expressions for the first and second Zagreb indices of graph operations containing the Cartesian product, composition, join, disjunction and symmetric difference of graphs will be presented.
242 citations
Cites background from "Wiener Index of Trees: Theory and A..."
...It is equal to the sum of distances between all pairs of vertices of the respective graph, see for details [3,4, 19]....
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01 Jan 2014
TL;DR: In this article, the authors present a survey on graphs extremal with respect to distance-based indices, with emphasis on the Wiener index, hyper-Wiener index and the Harary index.
Abstract: This survey outlines results on graphs extremal with respect to distance–based indices, with emphasis on the Wiener index, hyper–Wiener index, Harary index, Wiener polarity index, reciprocal complementary Wiener index, and terminal Wiener index.
210 citations
Cites background from "Wiener Index of Trees: Theory and A..."
...Then (1) ([32]) W (T ) ≤ W (Bn,Δ), with equality holding if and only if T ∼= Bn,Δ ....
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...Anyway, also in contemporary mathematical literature W (G) is usually referred to as the Wiener index [31,32, 41,53,96, 134,151] Wiener himself named WP (G) polarity number , and this quantity is nowadays usually -462-...
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References
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3,647 citations
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.
1,498 citations
"Wiener Index of Trees: Theory and A..." refers background in this paper
...If these eigenvalues are labelled so that λ1 λ2 ··· λn, then for all graphs λn = 0 whereas for all connected graphs λn−1 > 0. The theory of Laplacian spectra of graphs has been extensively studied (for a review see [ 95 , 96])....
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01 Jan 1991
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.
Abstract: The paper is essentially 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. Some new results and generalizations are added.
1,379 citations
"Wiener Index of Trees: Theory and A..." refers background in this paper
...If these eigenvalues are labelled so that λ1 λ2 ··· λn, then for all graphs λn = 0 whereas for all connected graphs λn−1 > 0. The theory of Laplacian spectra of graphs has been extensively studied (for a review see [95, 96 ])....
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...The following peculiar result was communicated around 1990 independently in several papers [93, 94, 96 , 97]....
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Book•
01 Jan 1986
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.
Abstract: A Chemistry and Topology.- 1 Topological Aspects in Chemistry.- 1.1 Topology in Chemistry.- 1.2 Abstraction in Science and How Far One Can Go.- 2 Molecular Topology.- 2.1 What is Molecular Topology?.- 2.2 Geometry, Symmetry, Topology.- 2.3 Definition of Molecular Topology.- B Chemistry and Graph Theory.- 3 Chemical Graphs.- 4 Fundamentals of Graph Theory.- 4.1 The Definition of a Graph.- 4.1.1 Relations.- 4.1.2 The First Definition of a Graph.- 4.1.3 The Second Definition of a Graph.- 4.1.4 Vertices and Edges.- 4.1.5 Isomorphic Graphs and Graph Automorphisms.- 4.1.6 Special Graphs.- 4.2 Subgraphs.- 4.2.1 Sachs Graphs.- 4.2.2 Matchings.- 4.3 Graph Spectral Theory.- 4.3.1 The Adjacency Matrix.- 4.3.2 The Spectrum of a Graph.- 4.3.3 The Sachs Theorem.- 4.3.4 The ?-Polynomial.- 4.4 Graph Operations.- 5 Graph Theory and Molecular Orbitals.- 6 Special Molecular Graphs.- 6.1 Acyclic Molecules.- 6.1.1 Trees.- 6.1.2 The Path and the Star.- 6.1.3 The Characteristic Polynomial of Trees.- 6.1.4 Trees with Greatest Number of Matchings.- 6.1.5 The Spectrum of the Path.- 6.2 The Cycle.- 6.3 Alternant Molecules.- 6.3.1 Bipartite Graphs.- 6.3.2 The Pairing Theorem.- 6.3.3 Some Consequences of the Pairing Theorem.- 6.4 Benzenoid Molecules.- 6.4.1 Benzenoid Graphs.- 6.4.2 The Characteristic Polynomial of Benzenoid Graphs.- 6.5 Hydrocarbons and Molecules with Heteroatoms.- 6.5.1 On the Question of the Molecular Graph.- 6.5.2 The Characteristic Polynomial of Weighted Graphs.- 6.5.3 Some Regularities in the Electronic Structure of Heteroconjugated Molecules.- C Chemistry and Group Theory.- 7 Fundamentals of Group Theory.- 7.1 The Symmetry Group of an Equilateral Triangle.- 7.2 Order, Classes and Representations of a Group.- 7.3 Reducible and Irreducible Representations.- 7.4 Characters and Reduction of a Reducible Representation.- 7.5 Subgroups and Sidegroups - Products of Groups.- 7.6 Abelian Groups.- 7.7 Abstract Groups and Group Isomorphism.- 8 Symmetry Groups.- 8.1 Notation of Symmetry Elements and Representations.- 8.2 Some Symmetry Groups.- 8.2.1 Rotation Groups.- 8.2.2 Groups with More than One n-Fold Axis, n > 2.- 8.2.3 Groups of Collinear Molecules.- 8.3 Transformation Properties and Direct Products of Irreducible Representations.- 8.3.1 Transformation Properties.- 8.3.2 Rules Concerning the Direct Product of Irreducible Representations.- 8.4 Some Applications of Symmetry Groups.- 8.4.1 Electric Dipole Moment.- 8.4.2 Polarizability.- 8.4.3 Motions of Atomic Nuclei: Translations, Rotations and Vibrations.- 8.4.4 Transition Probabilities for the Absorption of Light.- 8.4.5 Transition Probabilities in Raman Spectra.- 8.4.6 Group Theory and Quantum Chemistry.- 8.4.7 Orbital and State Correlations.- 9 Automorphism Groups.- 9.1 Automorphism of a Graph.- 9.2 The Automorphism Group A(G1).- 9.3 Cycle Structure of Permutations.- 9.4 Isomorphism of Graphs and of Automorphism Groups 112..- 9.5 Notation of some Permutation Groups.- 9.6 Direct Product and Wreath Product.- 9.7 The Representation of Automorphism Groups as Group Products.- 10 Some Interrelations between Symmetry and Automorphism Groups.- 10.1 The Idea of Rigid Molecules.- 10.2 Local Symmetries.- 10.3 Non-Rigid Molecules.- 10.4 What Determines the Respective Orders of the Symmetry and the Automorphism Group of a Given Molecule?.- D Special Topics.- 11 Topological Indices.- 11.1 Indices Based on the Distance Matrix.- 11.1.1 The Wiener Number and Related Quantities.- 11.1.2 Applications of the Wiener Number.- 11.2 Hosoya's Topological Index.- 11.2.1 Definition and Chemical Applications of Hosoya's Index.- 11.2.2 Mathematical Properties of Hosoya's Index.- 11.2.3 Example: Hosoya's Index of the Path and the Cycle.- 11.2.4 Some Inequalities for Hosoya's Index.- 12 Thermodynamic Stability of Conjugated Molecules.- 12.1 Total ?-Electron Energy and Thermodynamic Stability of Conjugated Molecules.- 12.2 Total ?-Electron Energy and Molecular Topology.- 12.3 The Energy of a Graph.- 12.4 The Coulson Integral Formula.- 12.5 Some Further Applications of the Coulson Integral Formula.- 12.6 Bounds for Total ?-Electron Energy.- 12.7 More on the McClelland Formula.- 12.8 Conclusion: Factors Determining the Total ?-Electron Energy.- 12.9 Use of Total ?-Electron Energy in Chemistry.- 13 Topological Effect on Molecular Orbitals.- 13.1 Topologically Related Isomers.- 13.2 Interlacing Rule.- 13.3 PE Spectra of Topomers.- 13.4 TEMO and a-Electron Systems.- 13.5 TEMO and Symmetry.- Appendices.- Appendix 1 Matrices.- Appendix 2 Determinants.- Appendix 3 Eigenvalues and Eigenvectors.- Appendix 4 Polynomials.- Appendix 5 Characters of Irreducible Representations of Symmetry Groups.- Appendix 6 The Symbols Used.- Literature.- References.
1,283 citations
Book•
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01 Jan 1990
1,185 citations
"Wiener Index of Trees: Theory and A..." refers background in this paper
...In the mathematical literature W seems to be first studied only in 1976 [32]; for a long time mathematicians were unaware of the (earlier) work on W done in chemistry (cf. the book [ 6 ])....
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...The Doyle–Graver formula holds not only for trees, but for all geodetic graphs [ 6 ]....
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