# Solved and Unsolved Problems of Structural Chemistry

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## Book Description

**Solved and Unsolved Problems of Structural Chemistry** introduces new methods and approaches for solving problems related to molecular structure. It includes numerous subjects such as aromaticity—one of the central themes of chemistry—and topics from bioinformatics such as graphical and numerical characterization of DNA, proteins, and proteomes. It also outlines the construction of novel tools using techniques from discrete mathematics, particularly graph theory, which allowed problems to be solved that many had considered unsolvable.

The book discusses a number of important problems in chemistry that have not been fully understood or fully appreciated, such as the notion of aromaticity and conjugated circuits, the generalized Hückel 4n + 2 Rule, and the nature of quantitative structure–property–activity relationships (QSARs), which have resulted in only partially solved problems and approximated solutions that are inadequate. It also describes advantages of mathematical descriptors in QSAR, including their use in screening combinatorial libraries to search for structures with high similarity to the target compounds.

Selected problems that this book addresses include:

- Multiple regression analysis (MRA)
- Insufficient use of partial ordering in chemistry
- The role of Kekulé valence structures
- The problem of protein and DNA alignment

**Solved and Unsolved Problems of Structural Chemistry** collects results that were once scattered in scientific literature into a thoughtful and compact volume. It sheds light on numerous problems in chemistry, including ones that appeared to have been solved but were actually only partially solved. Most importantly, it shows more complete solutions as well as methods and approaches that can lead to actualization of further solutions to problems in chemistry.

## Table of Contents

**Introduction**Problems of Antiquity

Famous Problems

Illustrious Problems

Hilbert Problems

P versus NP

On Unsolved Problems

References and Notes

**Mathematical Chemistry**

Discrete Mathematics and Chemistry

Partial Order

Enumeration of Kekulé Valence Structures

Graph Theory

Chemical Nomenclature

Molecular Descriptors

Canonical Labeling of Vertices

Counting of Paths

Experimental Mathematics

References and Notes

**Graph Theory and Chemistry**

Unfamiliarity with Graph Theory

On HMO, the Hückel Molecular Orbital Theory

The Story of Isospectral Graphs

The Difference in HMO and PPP Bond Orders

Graphene Edge

References and Notes

**Characteristic Polynomial**

On Characteristic Polynomial

On Construction of the Characteristic Polynomial

Fermat’s Last Theorem

Common Eigenvalues

Chemical Mathematics

Revisiting the Characteristic Polynomial

On Factoring of Characteristic Polynomial

Symmetric Function and Newton Identities

Summary

References and Notes

**Structure–Activity**

Two Cultures

Qualitative Regression Analysis

QSAR versus "QSAR"

On Quantitative Characterization of Molecular Structure

The Criticism That Is Not

Highly Collinear Descriptors: The Problem That Is Not

The Single Equation Fallacy

References and Notes

**Molecular Descriptors**

Structure Descriptors

The Connectivity Index

Topological Indices

Complex Biomolecular Networks and the Connectivity Index

^{1}χ Joseph Boscovic

Desiderata for Molecular Descriptors

Higher-Order Connectivity Indices

Molecular ID Numbers

Recent Modification of the Higher-Order Connectivity Indices

Variations of the Connectivity Index

On Characterization of Hetero-Atoms

The Variable Connectivity Index

Surprise, Surprise

The Structure–Structure Space

References and Notes

**Partial Ordering**

Searching for Regularities in Structure–Property Data

On Characterization of Molecular Structures

Partial Ordering

^{13}C Chemical Shifts Sums in Alkanes

Study of Proteome Maps Using Partial Ordering

References and Notes

**Novel Molecular Matrices**

Novel Molecular Descriptors

Novel Matrices in Chemistry

Generalized Distance Matrices

Ring Closure Matrix

Spanning Trees

Canonical Labels for Graphs

Canonical Labels for Adamantane

Walk Above Code for Polycyclic Graphs

Ring Closure Matrix of Adamantane

References and Notes

**On Highly Similar Molecules**

Search for Structurally Similar Molecules

Molecular Descriptors for Screening Combinatorial Libraries

Generalized Connectivity Indices

Connectivity Index χ(HD)

References and Notes

**Aromaticity Revisited**

Conjugated Circuits

Resonance Energies

In Search of a Generalized Hückel 4

*n*+ 2 Rule

On Classification of Polycyclic Conjugated Compounds

References and Notes

**Clar Aromatic Sextet**

Introduction

Qualitative Clar Aromatic Sextet Theory

Clar Aromatic Sextet Revisited

Pauling Bond Orders

Ring Bond Orders

Quantitative Aromatic Sextet Theory

Resonance Graphs

References and Notes

**Renormalization in Chemistry**

Continuum versus Discrete Models

Ring Currents via the Conjugated Circuits Model

Normalization of Ring Currents

Renormalization of Ring Currents

Concluding Remarks

Anecdotes

Plagiarism That Is Not

References and Notes

**Graphical Bioinformatics**

Characterization versus Indexing

Milestones of Graphical Bioinformatics

Spectral Representation of DNA

Four-Color 2-D Maps of DNA

Similarities and Dissimilarities among the DNA Sequences

Is a Four-Color DNA Map a Useful Novelty?

Amino Acid Adjacency Matrix

Decagonal Isometric Matrix

On the Sequence Alignment Problem

Amino Acid Sequential Matrix

Computer Program for the Exact Solution of the Protein Alignment Problem

References and Notes

**Beauties and Sleeping Beauties in Science**

Sleeping Beauties Revisited

On Beauty in Science

Five Sleeping Beauties

Aromaticity and Conjugation

4

*n*+ 2 and 4

*n*Rule for Polycyclic Conjugated Systems

Orthogonal Molecular Descriptors

Variable Connectivity Index

Proteome Maps: Order or Chaos?

Graphical Alignment of Proteins

Sleeping Giants

Summary

Apology

Acknowledgments

References and Notes

**Appendix 1: Early Hostility toward Chemical Graph Theory**

Appendix 2: Editorial Alert to Reviewers on Graph Theoretical Manuscripts

Appendix 3: Letter Relating to the Calculation of the Symmetry of a Graph

Appendix 4: On Detection of the Illegal Deck of Cards for Graph Reconstruction

Appendix 5: Solution to Halmos Handshake Problem

Appendix 6: The First Page of the Book of Euclid

Appendix 7: Full Quote of Immanuel Kant

Appendix 8: Construction of Canonical Labels for Diamantane

Appendix 9: Twelve Different Canonical Labelings of Diamantane

Appendix 10: The Eigenvalues and Eigenvectors of Hexatriene

Appendix 11: The Coefficients of All Antisymmetrical Molecular Orbitals of the Tetracene Derivative C

Appendix 13: "News and Views" on Chemical Topology from

Appendix 14: Conjugated Circuits for the Remaining Five Symmetry Nonequivalent Kekulé Valence Structures of Coronene

Appendix 15: Hostile Reports of Anonymous Referees on Novelty in Graphical Bioinformatics

Appendix 16: The Computer Program for Exact Pairwise Protein Alignment

Appendix 17: An Input and Output of the Program VESPA

Appendix 18: List of Problems of Structural Chemistry Presented in This Book

Appendix 19: Rigorous Definitions and Descriptions of a Selection of Mathematical Concepts of Discrete Mathematics

Appendix 2: Editorial Alert to Reviewers on Graph Theoretical Manuscripts

Appendix 3: Letter Relating to the Calculation of the Symmetry of a Graph

Appendix 4: On Detection of the Illegal Deck of Cards for Graph Reconstruction

Appendix 5: Solution to Halmos Handshake Problem

Appendix 6: The First Page of the Book of Euclid

Appendix 7: Full Quote of Immanuel Kant

Appendix 8: Construction of Canonical Labels for Diamantane

Appendix 9: Twelve Different Canonical Labelings of Diamantane

Appendix 10: The Eigenvalues and Eigenvectors of Hexatriene

Appendix 11: The Coefficients of All Antisymmetrical Molecular Orbitals of the Tetracene Derivative C

_{20}H_{14}Appendix 12: Halogen Compounds and Their Boiling PointsAppendix 13: "News and Views" on Chemical Topology from

*Nature*, November 15, 1974Appendix 14: Conjugated Circuits for the Remaining Five Symmetry Nonequivalent Kekulé Valence Structures of Coronene

Appendix 15: Hostile Reports of Anonymous Referees on Novelty in Graphical Bioinformatics

Appendix 16: The Computer Program for Exact Pairwise Protein Alignment

Appendix 17: An Input and Output of the Program VESPA

Appendix 18: List of Problems of Structural Chemistry Presented in This Book

Appendix 19: Rigorous Definitions and Descriptions of a Selection of Mathematical Concepts of Discrete Mathematics

## Author(s)

### Biography

**Milan Randić** earned his PhD from the University of Cambridge, England. He founded the Theoretical Chemistry group at the Institute Rudjer Bošković in Zagreb, Croatia, in 1960. He joined the department of chemistry at the University of Zagreb in 1965 before leaving for the United States. He later joined the department of mathematics and computer science at Drake University, Des Moines, Iowa, until his retirement in 1999 as a distinguished professor. He has won several awards and pays annual visits to the Laboratory of Chemometrics, National Institute of Chemistry, Ljubljana, Slovenia. He is a member of the Croatian Academy of Sciences and Arts, an honorary member of the Slovenian Chemical Society, and an honorary member of the National Institute of Chemistry in Ljubljana, from which he received the Grand Pregel Award in 2010. In 2014 he reported the exact solution of the protein alignment problem, which had existed for 45 years.

**Marjana Novic **is head of the Laboratory of Chemometrics at the National Institute of Chemistry, Ljubljana, Slovenia, and** **teaches chemometrics at the University of Ljubljana, where she earned her PhD from the Faculty of Chemistry and Chemical Technology in 1985. She started her career at the National Institute of Chemistry in Ljubljana, initially developing automated information systems for infrared and NMR spectroscopy. Her expertise includes the development of chemometrics methods, QSAR and ANN modeling, structural elucidation of transmembrane segments of membrane proteins, and innovative merging of chemometrics methods with molecular modeling, which facilitates effective drug design.

**Dejan Plavšić **is a senior research scientist at the NMR Center at the Rudjer Bošković Institute, Zagreb, Croatia. He earned his PhD in chemistry from the University of Zagreb. His research interests are in mathematical chemistry, chemical graph theory and its applications, metal clusters, organometallic compounds, and catalysis. Since 2003 he has been interested in extending graphical and numerical characterizations to DNA, proteins, and proteome maps using graph theoretical tools, entering thus into bioinformatics. Two of his papers in this area have received recognition as being among the 50 Most Cited Papers in Chemical Physics Letters during the 2003–2007 period. He is also one of the initial founding members of the International Academy of Mathematical Chemistry. In 2006 he was the first recipient of the International Award Latium between Europe and the Mediterranean for Medicine, Physics, or Chemistry.

## Reviews

"Already in the

Preface(on page XIII) the authors declare that their book is 'unusual and unconventional'. Indeed it is! Contrary to many other existing monographs concerned with chemical graph theory or its selected special topics, this book is full of interesting, unusual, and surprising excursions, making its reading a great pleasure. Everywhere in the book, we find historical details and anecdotes, related to physics, chemistry, physical chemistry, and mathematics, often coming from Milan Randic's personal experience. Colleagues interested in philosophy of science (or philosophy in general) will also find a wealth of intriguing ideas. For these colleagues we especially recommend Chapter 14, and its sections 'On Beauty in Science'and 'Sleeping Giants'. The same chapter will be a gold mine for scholars interested in the history of chemistry . . .

In summary, 'Solved and Unsolved Problems of Structural Chemistry'is a valuable treatise, outlining practically all facets of the research of Milan Randic in various areas of chemical graph theory. ... this book should be a valuable item in the personal library of all those who ever did any work or ever had any interest in chemical graph theory."

–Ivan Gutman,Match: Communications in Mathematical and in Computer Chemistry, 2016.

"The book is foremost readable, copiously and suitably illustrated, and set out in a logical manner… I would recommend this book to someone interested in chemical graph theory with any level of knowledge of physical chemistry."

–Colin F. Poole forChromatographia(2016) 79:1049–1050. DOI 10.1007/s10337-016-3110-4.