Ivan Gutman

Bio: Ivan Gutman is an academic researcher from University of Kragujevac. The author has contributed to research in topics: Wiener index & Topological index. The author has an hindex of 72, co-authored 963 publications receiving 30310 citations. Previous affiliations of Ivan Gutman include Vanderbilt University & University of Antioquia.

Mathematical concepts in organic chemistry

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.

Wiener Index of Trees: Theory and Applications

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.

Graph theory and molecular orbitals

Abstract: In the present chapter, as well as throughout the entire book, we assume that the reader knows the basic facts about the Huckel molecular orbital (HMO) theory [35, 51, 62]. Hence HMO theory is an approximate quantum-mechanical approach to the description of the π-electrons in unsaturated conjugated molecules. The wave function for a π-electron is presented in the LCAO form $$ {\psi_i} = \sum\limits_{{j = 1}}^n {{c_{{ij}}}} \left| {{p_j} >} \right. $$ (1) where {p j > symbolizes a p π -vorbital located on the j-th atom of the conjugated molecule, and the summation goes over all n atoms which participate in the conjugation.

Graph theory and molecular orbitals. XII. Acyclic polyenes

Abstract: A graph‐theoretical study of acyclic polyenes is carried out with an emphasis on the influence of branching on several molecular properties. A definition of branching is given and several branching indices are analyzed. The case of polyenes without a Kekule structure is discussed briefly. The main conclusions are: (a) thermodynamic stability of conjugated polyenes decreases with branching, but (b) reactivity, in general, increases with branching.

A tutorial on spectral clustering

Abstract: In recent years, spectral clustering has become one of the most popular modern clustering algorithms. It is simple to implement, can be solved efficiently by standard linear algebra software, and very often outperforms traditional clustering algorithms such as the k-means algorithm. On the first glance spectral clustering appears slightly mysterious, and it is not obvious to see why it works at all and what it really does. The goal of this tutorial is to give some intuition on those questions. We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. Advantages and disadvantages of the different spectral clustering algorithms are discussed.

Random graphs

Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

SMILES, a chemical language and information system. 1. introduction to methodology and encoding rules

Abstract: (1) Klamer, A. D. “Some Results Concerning Polyominoes”. Fibonacci Q. 1965, 3(1), 9-20. (2) Golomb, S. W. Polyominoes·, Scribner, New York, 1965. (3) Harary, F.; Read, R. C. “The Enumeration of Tree-like Polyhexes”. Proc. Edinburgh Math. Soc. 1970, 17, 1-14. (4) Lunnon, W. F. “Counting Polyominoes” in Computers in Number Theory·, Academic: London, 1971; pp 347-372. (5) Lunnon, W. F. “Counting Hexagonal and Triangular Polyominoes”. Graph Theory Comput. 1972, 87-100. (6) Brunvoll, J.; Cyvin, S. J.; Cyvin, B. N. “Enumeration and Classification of Benzenoid Hydrocarbons”. J. Comput. Chem. 1987, 8, 189-197. (7) Balaban, A. T., et al. “Enumeration of Benzenoid and Coronoid Hydrocarbons”. Z. Naturforsch., A: Phys., Phys. Chem., Kosmophys. 1987, 42A, 863-870. (8) Gutman, I. “Topological Properties of Benzenoid Systems”. Bull. Soc. Chim., Beograd 1982, 47, 453-471. (9) Gutman, I.; Polansky, O. E. Mathematical Concepts in Organic Chemistry·, Springer: Berlin, 1986. (10) To3i6, R.; Doroslovacki, R.; Gutman, I. “Topological Properties of Benzenoid Systems—The Boundary Code”. MATCH 1986, No. 19, 219-228. (11) Doroslovacki, R.; ToSic, R. “A Characterization of Hexagonal Systems”. Rev. Res. Fac. Sci.-Univ. Novi Sad, Math. Ser. 1984,14(2) 201-209. (12) Knop, J. V.; Szymanski, K.; Trinajstic, N. “Computer Enumeration of Substituted Polyhexes”. Comput. Chem. 1984, 8(2), 107-115. (13) Stojmenovic, L; Tosió, R.; Doroslovaóki, R. “Generating and Counting Hexagonal Systems”. Proc. Yugosl. Semin. Graph Theory, 6th, Dubrovnik 1985; pp 189-198. (14) Doroslovaóki, R.; Stojmenovió, I.; Tosió, R. “Generating and Counting Triangular Systems”. BIT 1987, 27, 18-24. (15) Knop, J. V.; Miller, W. R.; Szymanski, K.; Trinajstic, N. Computer Generation of Certain Classes of Molecules·, Association of Chemists and Technologists of Croatia: Zagreb, 1985.