Solved and Unsolved Problems of Structural Chemistry  book cover
1st Edition

Solved and Unsolved Problems of Structural Chemistry

ISBN 9780367862268
Published March 31, 2021 by CRC Press
492 Pages 180 B/W Illustrations

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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

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
References and Notes

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 1χ 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
13C 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 4n + 2 Rule
On Classification of Polycyclic Conjugated Compounds
References and Notes

Clar Aromatic Sextet
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
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
4n + 2 and 4n Rule for Polycyclic Conjugated Systems
Orthogonal Molecular Descriptors
Variable Connectivity Index
Proteome Maps: Order or Chaos?
Graphical Alignment of Proteins
Sleeping Giants
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 C20H14

Appendix 12: Halogen Compounds and Their Boiling Points

Appendix 13: "News and Views" on Chemical Topology from Nature, November 15, 1974

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

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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.


"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 for Chromatographia (2016) 79:1049–1050. DOI 10.1007/s10337-016-3110-4.