On applications of the retracing method for distance-regular graphs - Vencer, Norma Luz C. (Animo Repository, 2006)
This paper is an exposition of the article written by Akira Hiraki entitled Applications of Retracing Method for Distance-Regular Graphs published in European Journal of Combinatorics, April 2004 whose main results are as follows: Theorem 1.1 Let be a distance-regular graph of diameter d with r = |{ i |(ci, ai, bi) = (c1, a1, b1)}| 2 and cr+1 2. Let m, s and t be positive integers with s m, m + t d and (s, t) 6= (1,1). Suppose bms+1 = · · · = bm = 1 + bm+1, cm+1 = · · · = cm+t = 1 + cm and ams+2 = · · · = am+t1 = 0. Then the following hold. (1) If bm+1 2, then t r 2 bs/3c . (2) If cm 2, then s r 2 bt/3c . Corollary 1.2. Under the assumption of Theorem 1.1, the following hold. (1) If r = t and bm+1 2, then s 2. (2) If r = s and cm 2, then t 2. Corollary 1.3. Let be a distance-regular graph of valency k 3 with c1 = · · · = cr = 1, cr+1 = · · · = cr+t = 2 and a1 = · · · = ar+t1 = 0. 4 (1) If k 4, then t r 2 br/3c . (2) If 2 t = r, then is either the Odd graph, or the doubled Odd graph. (3) If 2 t = r 1, then is the Foster graph. This paper is an exposition of the article written by Akira Hiraki entitled Applications of Retracing Method for Distance-Regular Graphs published in European Journal of Combinatorics, April 2004 whose main results are as follows: Theorem 1.1 Let be a distance-regular graph of diameter d with r = |{ i |(ci, ai, bi) = (c1, a1, b1)}| 2 and cr+1 2. Let m, s and t be positive integers with s m, m + t d and (s, t) 6= (1,1). Suppose bms+1 = · · · = bm = 1 + bm+1, cm+1 = · · · = cm+t = 1 + cm and ams+2 = · · · = am+t1 = 0. Then the following hold. (1) If bm+1 2, then t r 2 bs/3c . (2) If cm 2, then s r 2 bt/3c . Corollary 1.2. Under the assumption of Theorem 1.1, the following hold. (1) If r = t and bm+1 2, then s 2. (2) If r = s and cm 2, then t 2. Corollary 1.3. Let be a distance-regular graph of valency k 3 with c1 = · · · = cr = 1, cr+1 = · · · = cr+t = 2 and a1 = · · · = ar+t1 = 0. 4 (1) If k 4, then t r 2 br/3c . (2) If 2 t = r, then is either the Odd graph, or the doubled Odd graph. (3) If 2 t = r 1, then is the Foster graph.