NDLI: Parallelogram-free distance-regular graphs having completely regular strongly regular subgraphs

Content Provider	Springer Nature Link
Author	Suzuki, Hiroshi
Copyright Year	2009
Abstract	Let Γ=(X,R) be a distance-regular graph of diameter d. A parallelogram of length i is a 4-tuple xyzw consisting of vertices of Γ such that ∂(x,y)=∂(z,w)=1, ∂(x,z)=i, and ∂(x,w)=∂(y,w)=∂(y,z)=i−1. A subset Y of X is said to be a completely regular code if the numbers $$\pi_{i,j}=\|\Gamma_{j}(x)\cap Y\|\quad (i,j\in \{0,1,\ldots,d\})$$ depend only on i=∂(x,Y) and j. A subset Y of X is said to be strongly closed if $$\{x\mid \partial(u,x)\leq \partial(u,v),\partial(v,x)=1\}\subset Y,\mbox{ whenever }u,v\in Y.$$ Hamming graphs and dual polar graphs have strongly closed completely regular codes. In this paper, we study parallelogram-free distance-regular graphs having strongly closed completely regular codes. Let Γ be a parallelogram-free distance-regular graph of diameter d≥4 such that every strongly closed subgraph of diameter two is completely regular. We show that Γ has a strongly closed subgraph of diameter d−1 isomorphic to a Hamming graph or a dual polar graph. Moreover if the covering radius of the strongly closed subgraph of diameter two is d−2, Γ itself is isomorphic to a Hamming graph or a dual polar graph. We also give an algebraic characterization of the case when the covering radius is d−2.
Starting Page	401
Ending Page	413
Page Count	13
File Format	PDF
ISSN	09259899
Journal	Journal of Algebraic Combinatorics
Volume Number	30
Issue Number	3
e-ISSN	15729192
Language	English
Publisher	Springer US
Publisher Date	2009-02-04
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Distance-regular graph Association scheme Homogeneity Completely regular code Group Theory and Generalizations Computer Science Order, Lattices, Ordered Algebraic Structures Convex and Discrete Geometry Combinatorics
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Algebra and Number Theory

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