Perfect 2‐colorings of Hamming graphs

The Norton product is defined on each eigenspace of a distance regular graph by the orthogonal projection of the entry-wise product. The resulting algebra, known as the Norton algebra, is a commutative nonassociative algebra that is useful in group theory due to its interesting automorphism group. We provide a formula for the Norton product on each eigenspace of a Hamming graph using linear characters. We construct a large subgroup of automorphisms of the Norton algebra of a Hamming graph and completely describe the automorphism group in some cases. We also show that the Norton product on each eigenspace of a Hamming graph is as nonassociative as possible, except for some special cases in which it is either associative or equally as nonassociative as the so-called double minus operation previously studied by the author, Mickey, and Xu. Our results restrict to the hypercubes and extend to the halved and/or folded cubes, the bilinear forms graphs, and more generally, all Cayley graphs of finite abelian groups.

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The Minimum Stretch Spanning Tree Problem for Hamming Graphs and Higher-Dimensional Grids

Journal of Interconnection Networks ◽

10.1142/s0219265920500048 ◽

2020 ◽

Vol 20 (01) ◽

pp. 2050004

Author(s):

LAN LIN ◽

YIXUN LIN

Keyword(s):

Communication Networks ◽

Spanning Tree ◽

Optimization Problem ◽

Maximum Distance ◽

Np Hard ◽

Graph Invariant ◽

Minimum Value ◽

Higher Dimensional ◽

Hamming Graphs

The minimum stretch spanning tree problem for a graph G is to find a spanning tree T of G such that the maximum distance in T between two adjacent vertices is minimized. The minimum value of this optimization problem gives rise to a graph invariant σ(G), called the tree-stretch of G. The problem has been proved NP-hard. In this paper we present a general approach to determine the exact values σ(G) for a series of typical graphs arising from communication networks, such as Hamming graphs and higher-dimensional grids (including hypercubes).

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The retracts of Hamming graphs

Discrete Mathematics ◽

10.1016/0012-365x(92)90054-j ◽

1992 ◽

Vol 102 (2) ◽

pp. 197-218 ◽

Cited By ~ 21

Author(s):

Elke Wilkeit

Keyword(s):

Hamming Graphs

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EMBEDDING PERMUTATION GROUPS INTO WREATH PRODUCTS IN PRODUCT ACTION

Journal of the Australian Mathematical Society ◽

10.1017/s1446788712000110 ◽

2012 ◽

Vol 92 (1) ◽

pp. 127-136 ◽

Cited By ~ 2

Author(s):

CHERYL E. PRAEGER ◽

CSABA SCHNEIDER

Keyword(s):

Permutation Group ◽

Wreath Product ◽

Automorphism Groups ◽

Wreath Products ◽

Error Correcting Codes ◽

Permutation Groups ◽

Graph Products ◽

Product Action ◽

Hamming Graphs

AbstractWe consider the wreath product of two permutation groups G≤Sym Γ and H≤Sym Δ as a permutation group acting on the set Π of functions from Δ to Γ. Such groups play an important role in the O’Nan–Scott theory of permutation groups and they also arise as automorphism groups of graph products and codes. Let X be a subgroup of Sym Γ≀Sym Δ. Our main result is that, in a suitable conjugate of X, the subgroup of SymΓ induced by a stabiliser of a coordinate δ∈Δ only depends on the orbit of δ under the induced action of X on Δ. Hence, if X is transitive on Δ, then X can be embedded into the wreath product of the permutation group induced by the stabiliser Xδ on Γ and the permutation group induced by X on Δ. We use this result to describe the case where X is intransitive on Δ and offer an application to error-correcting codes in Hamming graphs.

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