Date of Publication

2006

Document Type

Master's Thesis

Degree Name

Master of Science in Mathematics

Subject Categories

Mathematics

College

College of Science

Department/Unit

Mathematics and Statistics

Thesis Adviser

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.

Abstract Format

html

Language

English

Format

Electronic

Accession Number

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

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