Distance-regular subgraphs in a distance-regular graph. I
type: Item
instance of: scholarly article
title: Distance-regular subgraphs in a distance-regular graph. I
zbMATH Open document ID: 0836.05080
DOI: 10.1016/0195-6698(95)90040-3
author: Akira Hiraki
published in: European Journal of Combinatorics
publication date: 1996-04-22T00:00:00Z
review text: Let \(\Gamma\) be a distance-regular graph with intersection array \(\{b_0, \dots, b_{d - 1}; c_1, \dots, c_d\}\). Then \(\Gamma\) is regular with \(k = b_0\). Set \(a_j = k_j - b_j - c_j\), \(l(\alpha, \beta, \gamma) = |\{j \mid (c_j, a_j, b_j) = (\alpha, \beta, \gamma) \} |\). It is known that a distance-regular graph with \(r = l(1, a_1, b_1) = 1\) and \(c_3 = 1\) contains a distance- regular subgraph of diameter 2 with \(c_2 = 1\). The author generalizes this result to general \(r\), and this paper considers the case \(a_1 = 0\). Theorem. Let \(\Gamma\) be a distance-regular graph with \(r = l(1,0,k - 1) > 0\). If \(c_{2r + 1} = 1\) then \(\Gamma\) contains a Moore subgraph of valency \(a_{r + 1} + 1\) and diameter \(r + 1\) (in particular, \(r = 1\) or \(a_{r + 1} = 1)\).
Mathematics Subject Classification ID: 05E30
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 823418
zbMATH Keywords: Moore graph
zbMATH Keywords: distance-regular graph
zbMATH Keywords: intersection array
zbMATH Keywords: distance-regular subgraph
reviewed by: Alexandre A. Makhnev
MaRDI profile type: Publication
full work available at URL: 0195-6698(95)90040-3
OpenAlex ID: W4205657677
cites work: Q3218140
cites work: A remark on the intersection arrays of distance-regular graphs
cites work: Q3992965
cites work: A constant bound on the number of columns \((1,k-2,1)\) in the intersection array of a distance-regular graph
cites work: CIRCUIT CHASING TECHNIQUE FOR A DISTANCE-REGULAR GRAPH WITH c 2r+1=1
cites work: Distance-regular subgraphs in a distance-regular graph. II
description: scientific article; zbMATH DE number 823418