## Master's Theses

2006

Master's Thesis

#### Degree Name

Master of Science in Mathematics

Mathematics

#### College

College of Science

#### Department/Unit

Mathematics and Statistics Department

Severino D. Diesto

#### Defense Panel Chair

Arlene A. Pascasio

#### Defense Panel Member

Severino V. Gervacio
Erminda C. Fortes

#### Abstract/Summary

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.

html

English

Electronic

CDTG004168

#### Shelf Location

Archives, The Learning Commons, 12F Henry Sy Sr. Hall

#### Physical Description

vi, 76 leaves, 28 cm. ; Typescript

#### Keywords

Graph theory; Theory of graphs; Distance-regular graphs

wf_yes

COinS