Chemical applications of discrete mathematics and graph theory are briefly reviewed, including philosophical implications. Using the concept of dualist (inner graph-theoretical duals) it was possible to classify (cata-peri-corona classes) and enumerate benzenoid and diamondoid hydrocarbons. By associating numbers with molecular graphs, one can use these numbers (topological indices) for correlations with properties of chemical compounds − an early, simple, and rapid approach to drug design. The two types of atoms (metals and non-metals), are connected by three types pf chemical bonds (ionic, metallic, and covalent) that lead to four types pf lattices (ionic, metallic, atomic and molecular), allowing quick "2-3-4 grasp" of chemistry. The Periodic System of Elements, which is the cornerstone of chemistry and atomic physics, is in danger of being presented wrongly, devoid of the symmetry based on electronic s, p, d, and f shells; several possibilities for showing correctly these shells are discussed.
This article reviews results of research on the development of graph theoretical chemodescriptors, topological indices in particular, and proteomics as well as DNA/RNA sequence based biodescriptors and their applications in predicting property/bioactivity of chemicals as well as viruses. Use of biodescriptors in the characterization of emerging pathogens like the Zika virus (ZIKV) has been discussed. The use of proper statistical methods in model building is emphasized with special reference to research carried out by the author of this review.
In this article we have illustrated one solved problem in chemistry, which left room for improvements. This instance is raising an issue when a solved problem is to be considered solved. As we will see by using novel tools, not known at the time of solving a problem, one can arrived at additional unknown information on the problem. We will consider several problems of chemistry, some even considered as having no exact solution, which have been solved by using previously unknown concepts in chemistry. The last problem which we consider has been solved, but the problem is still open, as it might have additional solutions.
The personal history of the present author's research career of almost forty-five years in the application of graph theory to chemistry is briefly described in an essay with particular reference to the author's "bright side of mathematical chemistry." The chosen topics are as follows: two seminal papers by the present author, more than 1500 citations of his topological index, Z, Hosoya polynomial, Hosoya items, more than 200 papers carrying "Hosoya" in the title, Erdös number, etc.
Mathematical stereochemistry is discussed by surveying books written on Fujita's USCI (unit-subduced-cycle-index) approach, Fujita's proligand method, Fujita's stereoisogram approach, and related matters from a viewpoint of developing an interdisciplinary chemistry/mathematics field.
We examined the length-dependence of algebraic structure count (ASC) of ladder graphs. We derived the recursion formula for ASC of ladder graphs and found the periodicity in the length-dependence of ASC.By using this periodicity of ASC we found the same periodicities in non-bonding molecular orbitals and in HOMO-LUMO gaps of ladder graphs.
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