On applications of the retracing method for distance-regular graph

Abstract

This thesis 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. The main results of the article are as follows: \(\mathbf{Theorem \,1.1} \, \mathrm{Let}\, \Gamma \,\mathrm{be\,a\,distance-regular\,graph \,of \,diameter} \,d \,\mathrm{with}\)

\[r = |\{ i|(c_{i}, a_{i}, b_{i}) = (c_{1}, a_{1}, b_{1})\}| \geq 2\]

\(\mathrm{and} \, c_{r+1} \geq 2. \, \mathrm{Let} \, m, s \, \mathrm{and} \, t \, \mathrm{be\,positive\,integers\,with\,} s \leq \, m, m + t \leq d \, \mathrm{and} \, (s, t)\) \( \neq (1,1). \mathrm{Suppose} \, b_{m-s+1} = \cdots = b_{m} = 1 + b_{m+1}, c_{m+1} = \cdots = c_{m+t} = 1 + c_{m}\) \(\mathrm{and} \, a_{m-s+2} = \cdots = a_{m+t-1} = 0. \, \mathrm{Then\,the\,following\,hold.}\)

\[\mathrm{If}\,b_{m+1} ≥ 2,\,\mathrm{then} \,t ≤ r – 2 \lfloor\,s/3\,\rfloor. \tag{1}\] \[\mathrm{If} \,c_{m} ≥ 2, \mathrm{then} \,s ≤ r – 2 \lfloor\,t/3\,\rfloor. \tag{2}\]

\(\mathbf{Corollary \,1.2} \, \mathrm{Under\,the\,assumption\,of\,Theorem\,1.1,\,the\,following\,hold.}\)

\[\mathrm{If} \, r = t \, \mathrm{and} \, b_{m+1} \geq 2, \, \mathrm{then} \, s \leq 2. \tag{1}\] \[\mathrm{If} \, r = s \, \mathrm{and} \, c_{m} \geq 2, \, \mathrm{then} \, t \leq 2. \tag{2}\]

\(\mathbf{Corollary \,1.3.} \, \mathrm{Let} \, \Gamma \, \mathrm{be\,a\,distance-regular\,graph\,of\,valency} \, k \geq \, 3 \, \mathrm{with}\) \( c_{1} = \cdots = c_{r} = 1, c_{r+1} = \cdots = c_{r+t} = 2 \, \mathrm{and} \, a_{1} = \cdots = a_{r+t-1} = 0.\)

\[\mathrm{If} \, k \geq 4, \, \mathrm{then} \, t \leq r-2 \lfloor\,r/3\rfloor. \tag{1}\] \[\mathrm{If} \, 2 \leq t = r, \, \mathrm{then} \, \Gamma \, \mathrm{is \, either \, the \, Odd \, graph, \, or \, the \, doubled \, Odd \, graph.}\tag{2}\] \[\mathrm{If} \, 2 \leq t = r – 1, \, \mathrm{then} \, \Gamma \, \mathrm{is \, the \, Foster \, graph.} \tag{3}\]