On the girth of a thin distance-regular graph - Toto, Maria Criselda S. (Animo Repository, 2003)
This thesis is an exposition of the article The Girth of a Thin Distance-regular Graph by Benjamin V.C. Collins, which was published in the Journal of Graphs and Combinatorics (1997) [3]. Let t denote a distance-regular graph with vertex set X, diameter D greater than or equal to 3, and valency k greater than or equal to 3. For a fixed vertex x E X, let T(x) denote the Terwilliger algebra of t with respect to x. An irreducible T(x)-module W is thin if dim E*iW less than or equal to 1 (0 less than or equal to i less than or equal to D), where E*i(x) is the ith dual idempotent of t with respect to x. The graph t is called thin if every irreducible T(x)-module is thin with respect to every vertex x E X. Moreover, a regular generalized quadrangle is a bipartite distance-regular graph with girth 8 and diameter D = 4. The main results of [3] that are explained in detail in this thesis are as follows: t is a regular generalized quadrangle if and only if t is thin and the intersection number c3 = 1. Moreover, if t is thin then the girth is 3, 4, 6 or 8. The girth is exactly 8 when t is a regular generalized quadrangle.