This research focusses on the generalization of Professor Terwilliger's recent classification of the triply-regular graphs to arbitrary P- and Q-polynomial association schemes, development of the theory of thin distance-regular graphs, and further development of the Professor Terwilliger's theory of S and T algebras. Graphs appear throughout mathematics, making graph theory an important topic of study not only for its intrinsic interest, but for its many applications in mathematics and related topics. It serves as a tool in modeling many complex real world systems such as those encountered in computing and network systems.
