#### Term of Award

Spring 2014

#### Degree Name

Master of Science in Mathematics (M.S.)

#### Document Type and Release Option

Thesis (open access)

#### Department

Department of Mathematical Sciences

#### Committee Chair

Hua Wang

#### Committee Member 1

Amanda Stewart

#### Committee Member 2

Ionut Iacob

#### Abstract

Chemical graph theory began as a way for mathematicians to bring together the areas of the Physical Sciences and Mathematics. Through its use, mathematicians are able to model chemical systems, predict their properties as well as structure-property relationships. In this dissertation, we consider two questions involving chemical graph theory and its applications. We first look at tree-like polyphenyl systems, which form an important family of compounds in Chemistry, particularly in Material Science. The importance can be seen in LEDs, transmitters, and electronics. In recent years, many extremal results regarding such systems under specific constraints have been reported. More specifically are the sub-categories of such systems with extremal Wiener indices. We provide a labeling of the vertices on each hexagon, which facilitates the illustration of a tree-like polyphenyl system with its corresponding tree structure. This approach helps to characterize the extremal tree-like polyphenyl systems with respect to the Wiener index (ones that minimize or maximize the Wiener index). This method can also be used to order these systems and the results will aid in predicting the physical properties of compounds. We further compare the study between tree-like polyphenyl systems that resulted from different tree structures. We then focus on the application of a general weighted distance-based indexing system, similar to but more complicated than the Wiener index, to devise a way to determine the binding of proteins in biochemical systems. Proteins, composed of amino acids, are important biological molecules that dictate a wide range of functions on a cellular level. Although proteins have various basic sequences, proteins can be created to have similar functions. We compare peptide sequences that govern protein binding to methylated DNA. We do this by first devising an index for each amino acid and compare the overall index of a peptide sequence in order to characterize the binding capability between original proteins and mimics.

#### Recommended Citation

Williford, Tabitha N., "Structure vs. Properties Using Chemical Graph Theory" (2014). *Electronic Theses and Dissertations*. 1082.

https://digitalcommons.georgiasouthern.edu/etd/1082