On tight distance-regular graphs - Sinlao, Rowena A. (Animo Repository, 2000)
This thesis is an expository work taken from Sections 1-8 of the paper entitled Tight Distance-Regular Graphs by Aleksandar Jurisic, Jack Koolen, and Paul Terwilliger which will appear in the Journal of Algebraic Combinatorics. It focuses on the following result: Let T = (X, R) be a distance-regular graph with diameter d greater than or equal to 3 and eigenvalues k = Oo greater than O1 greater than ... greater than Od. Then the valency k, intersection numbers a1 and b1 satisfy (O1 + k over a1 + 1) (Od + k over a1 + 1) greater than or equal to -ka1b1 over (a1 + 1)2 T is said to be tight whenever T is nonbipartite, and the equality above holds. It also discusses characterizations of tight distance-regular graphs which involve intersection numbers and cosine sequences.