Mathematical Modeling for Ring Systems in Molecular Networks

Abstract

In computational mathematics, Graph Theory serves as an abstract model for chemical compounds.

Induced cycles, i.e. cycles with no chords, are referred to rings in molecules, and these rings have an important physical meaning in Chemistry. The mathematical analysis and development of algorithms for the ring perception problem is analogous to cycle detection in graph theory. In this work, we are interested in the representation of chemical structures using graphs and the detection of rings in these structures.

In the first chapter, we develop a polynomial time algorithm for the detection of all small induced cycles in a given graph $G$. We achieve a complexity of $\mathcal{O}(m^3n + n^2)$ for a graph of $m$ edges and $n$ vertices. Then, we apply this approach to several chemical compounds such as fullerenes, cata-condensed benzenoids, protein structures and others.

Many chemical properties of fullerenes and benzenoid systems can be explained in Mathematics in terms of the number of perfect matchings, the Clar number, the Fries number, the HOMO-LUMO energy gap, etc. These are some of predictors of molecules stability.

In the second chapter, we investigate the Fries number and Clar number for hexagonal systems and show that a cata-condensed hexagonal system has a maximum resonant set containing a maximum independent resonant set, which is conjectured for all hexagonal systems. Further, our computation results demonstrate that there exist many contra-pairs, and, for stability predictor of hexagonal systems, the Clar number is better than Fries number. Lastly, we compute the Clar number and Fries number of all isomers of fullerenes $C_{20}-C_{60}$ by using integer linear programming in addition to calculating the HOMO-LUMO energy gap of all fullerenes isomers.