Ivan Gutman

Wiener Index of Trees: Theory and Applications

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.

The Energy of a Graph: Old and New Results

Let G be a graph possessing n vertices and m edges. The energy of G, denoted by E = E(G), is the sum of the absolute values of the eigenvalues of G. The connection between E and the total electron energy of a class of organic molecules is briefly outlined. Some (known) fundamental mathematical results on E are presented: the relation between E(G) and the characteristic polynomial of G, lower and upper bounds for E, especially those depending on n and m, graphs extremal with respect to E, n-vertex graphs for which E(G) > E(K n ). The characterization of the n-vertex graph(s) with maximal value of E is an open problem.

Graph theory and molecular orbitals. XII. Acyclic polyenes

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.

Topological approach to the chemistry of conjugated molecules

1. Introduction.- 2. Graphs in Chemistry.- 2.1. Basic Definitions and Concepts of Graph Theory.- 2.1.1. Definition of a Graph.- 2.1.2. The Adjacency Matrix of a Graph.- 2.1.3. Isomorphism of the Graphs.- 2.1.4. Further Characterization of a Graph.- 2.2. Graphs and Topology.- 2.2.1. Path, Length and Distance.- 2.2.2. Neighbours. The Invariants of a Graph.- 2.2.3. Ring and Oriented Ring. Regular and Complete Graphs. The Ring and the Edge Components of a Graph.- 2.2.4. Sachs Graphs with N Vertices.- 2.3. Graphs Representing Conjugated Molecules.- 2.3.1. Planar Graphs. Colouring of Graphs.- 2.3.2. Huckel Graphs.- 2.3.3. Trees. Benzenoid Graphs.- 2.4. Graph Spectrum. Sachs Theorem.- 2.4.1. Graph Spectrum.- 2.4.2. Graph Spectral Properties of Particular Classes of Graphs.- 2.4.3. Sachs Theorem.- 2.5. Topology and Simple Molecular Orbital Model.- 2.6. Application of the Coulson-Sachs Graphical Method.- 2.7. Extension of Graph-Theoretical Considerations to Mobius Structures.- 3. Total Pi-Electron Energy.- 3.1. Introduction.- 3.2. Identities And Inequalities.- 3.2.1. The Fundamental Identity.- 3.2.2. Relations Between Epi The Adjacency Matrix and the Density Matrix.- 3.2.3. The Loop Rule.- 3.2.4. Inequalities for Epi.- 3.3. The Coulson Integral Formula.- 3.3.1. The First Integral Formula.- 3.3.2. Further Coulson-Type Formulas. I.- 3.3.3. An Application of the Coulson Integral Formula: The Tree with Maximal Energy.- 3.3.4. Further Coulson-Type Formulae. II.- 3.3.5. A Class of Approximate Topological Formulas for Epi.- 3.4. Topological Factors Determining the Gross Part of Epi.- 3.5. The Influence of Cycles: The Huckel Rule.- 3.5.1. General Considerations.- 3.5.2. The Huckel Rule.- 3.5.3 An Application: The Huckel Rule for Annulenes.- 3.5.4. Extension of the Huckel Rule to Nonalternant Systems.- 3.6. The Influence of KekulE Structures.- 3.6.1. Structure Count and Algebraic Structure Count.- 3.6.2. The Basic Postulate of Resonance Theory.- 3.7. The Influence of Branching.- 3.7.1. Violation of the Basic Postulate of Resonance Theory.- 3.8. Summary.- 4. Resonance Energy.- 4.1. Introduction.- 4.2. Classical and Dewar Resonance Energies.- 4.3. Topological Resonance Energy.- 4.3.1. The Mathematical Basis.- 4.3.2. The Computation of the Acyclic Polynomial.- 4.4. Tre as a Criterion of Aromatic Stability. Correlation with Experimental Findings.- 4.5. Concluding Remarks.- 5. Reactivity of Conjugated Structures.- 5.1. Localization Energy.- 5.2. Dewar Number.- 5.3. Topological Approach to Localization Energy.- 5.4. Topological Aspect of Dewar Number.- 5.5. Nonbonding Molecular Orbitals.- 6. Conclusions.- 7. Literature.

Variable neighborhood search for extremal graphs: IV: Chemical trees with extremal connectivity index

Abstract By means of the variable neighborhood search algorithm, a newly designed heuristic approach to combinatorial optimization, we established the structure of the chemical trees possessing extremal (maximal and minimal) values for the Randic connectivity index ( χ ). These findings were eventually corroborated by rigorous mathematical proofs. As could have been anticipated, the n -vertex tree with maximum χ is the path. The n -vertex chemical tree with minimum χ -value is not unique. The structures of such chemical trees (which should be considered as the graph representations of the most branched alkanes) are fully characterized.

Wiener Index of Hexagonal Systems

The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph. Hexagonal systems (HSs) 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 HSs extremal w.r.t. W, and on integers that cannot be the W-values of HSs. A few open problems are mentioned. The chemical applications of the results presented are explained in detail.

Acyclic systems with extremal Hckel ?-electron energy

The main result of the present work is the proof that among acyclic polyenes CnHn+2, the linear isomer H2C=(CH)n−2=CH2 has maximal HMO π-electron energy. The 1,1-divinyl isomer (H2C=CH)2C(CH)n−6=CH2 has maximal π-energy among branched acyclic systems. Among trees withn vertices, the star has minimal energy. A number of additional inequalities for HMO total π-electron energy of acyclic conjugated systems are proved.

On the theory of the matching polynomial

In this paper we report on the properties of the matching polynomial α(G) of a graph G. We present a number of recursion formulas for α(G), from which it follows that many families of orthogonal polynomials arise as matching polynomials of suitable families of graphs. We consider the relation between the matching and characteristic polynomials of a graph. Finally, we consider results which provide information on the zeros of α(G).

The Quasi-Wiener and the Kirchhoff Indices Coincide

In 1993 two novel distance-based topological indices were put forward. In the case of acyclic molecular graphs both are equal to the Wiener index, but both differ from it if the graphs contain cycles. One index is defined (Mohar, B.; Babic, D.; Trinajstic, N. J. Chem. Inf. Comput. Sci. 1993, 33, 153−154) in terms of eigenvalues of the Laplacian matrix, whereas the other is conceived (Klein, D. J.; Randic, M. J. Math. Chem. 1993, 12, 81−95) as the sum of resistances between all pairs of vertices, assuming that the molecule corresponds to an electrical network, in which the resistance between adjacent vertices is unity. Eventually, the former quantity was named quasi-Wiener index and the latter Kirchhoff index. We now demonstrate that the quasi-Wiener and Kirchhoff indices of all graphs coincide.

Variable Neighborhood Search for Extremal Graphs 2. Finding Graphs with Extremal Energy

The recently developed Variable Neighborhood Search (VNS) metaheuristic for combinatorial and global optimization is outlined together with its specialization to the problem of finding extremal graphs with respect to one or more invariants and the corresponding program (AGX). We illustrate the potential of the VNS algorithm on the example of energy E, a graph invariant which (in the case of molecular graphs of conjugated hydrocarbons) corresponds to the total π-electron energy. Novel lower and upper bounds for E are suggested by AGX and several conjectures concerning (molecular) graphs with extremal E values put forward. Moreover, most of the bounds are proved to hold.