Root systems and the Johnson and Hamming graphs
type: Item
instance of: scholarly article
title: Root systems and the Johnson and Hamming graphs
zbMATH Open document ID: 0614.05048
DOI: 10.1016/S0195-6698(87)80023-9
author: Paul Terwilliger
published in: European Journal of Combinatorics
publication date: 1987-01-01T00:00:00Z
review text: Let \(\Gamma\) be any distance regular graph with diameter d, \(c_ 2\geq 0\) and \[ c_ i-c_{i-1}+b_{i-1}-b_ i-a_ i-2=0. \] Then it is shown that \(\Gamma\) is a graph of Hamming type one of the Johnson graphs, the halved graph of the n-cube, a cocktail party graph, the Shrikhande graph or one of a finite number of exceptional graphs with \(d\leq 8.\) This result improves an earlier result of the author. The proof is built round the idea that any \(\Gamma\) satisfying the equation above can be represented by a set of equi-length vectors spanning a Euclidean space which also contains a root system consisting of an orthogonal union of root systems of type \(A_ n\), \(D_ n\), \(E_ 6\), \(E_ 7\), \(E_ 8\). The graphs listed above appear as consequences of the form of the root system.
Mathematics Subject Classification ID: 05C75
Mathematics Subject Classification ID: 05C99
zbMATH DE Number: 3993634
zbMATH Keywords: distance regular graph
zbMATH Keywords: root systems
MaRDI profile type: Publication
cites work: On the tetrahedral graph
cites work: Q3218140
cites work: A new 5‐arc‐transitive cubic graph
cites work: Automorphic graphs and the Krein condition
cites work: A characterization of tetrahedral graphs
cites work: There are only finitely many finite distance-transitive graphs of given valency greater than two
cites work: Line graphs, root systems, and elliptic geometry
cites work: Q3262641
cites work: Q3941436
cites work: The Uniqueness of the Triangular Association Scheme
cites work: Q5707657
cites work: A characterization of the Tm graph
cites work: Characterization of H(n,q) by the parameters
cites work: Q3942064
cites work: On the Uniqueness of the Triangular Association Scheme
cites work: Introduction to Lie Algebras and Representation Theory
cites work: On the uniqueness of the tetrahedral association schemes
cites work: Infinite distance transitive graphs of finite valency
cites work: Characterization of the odd graphs \(O_ k \)by parameters
cites work: Characterization of the graphs of the Johnson schemes G(3k,k) and G(3k+1,k)
cites work: On the uniqueness of the graphs G(n,k) of the Johnson schemes
cites work: On a Characterization of the Triangular Association Scheme
cites work: The Uniqueness of the $\mathrm{L}_2$ Association Scheme
cites work: The diameter of bipartite distance-regular graphs
cites work: Distance-regular graphs with girth 3 or 4: I
cites work: Distance-regular graphs and (s,c,a,k)-graphs
cites work: The Johnson graph \(J(d,r)\) is unique if \((d,r)\neq (2,8)\)
cites work: Q5550800
full work available at URL: s0195-6698(87)80023-9
OpenAlex ID: W2019781231
description: scientific article; zbMATH DE number 3993634