scholarly journals Eigenfunctions and minimum 1-perfect bitrades in the Hamming graph

2021 ◽  
Vol 344 (3) ◽  
pp. 112228
Author(s):  
Alexandr Valyuzhenich
Keyword(s):  
Hamming Graph

scholarly journals Norton Algebras of the Hamming Graphs via Linear Characters

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Jia Huang
Keyword(s):  
Group Theory ◽  
Automorphism Group ◽  
Nonassociative Algebra ◽  
Bilinear Forms ◽  
Distance Regular Graph ◽  
Finite Abelian Groups ◽  
Hamming Graph ◽  
Large Subgroup ◽  
Special Cases ◽  
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.

Cayley graphs of graded rings

2018 ◽  
Vol 17 (06) ◽  
pp. 1850116
Author(s):  
Saadoun Mahmoudi ◽  
Shahram Mehry ◽  
Reza Safakish
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Graded Rings ◽  
Graded Ring ◽  
Total Graph ◽  
Hamming Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Cayley Sum Graph

Let [Formula: see text] be a subset of a commutative graded ring [Formula: see text]. The Cayley graph [Formula: see text] is a graph whose vertex set is [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. The Cayley sum graph [Formula: see text] is a graph whose vertex set is [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Let [Formula: see text] be the set of homogeneous elements and [Formula: see text] be the set of zero-divisors of [Formula: see text]. In this paper, we study [Formula: see text] (total graph) and [Formula: see text]. In particular, if [Formula: see text] is an Artinian graded ring, we show that [Formula: see text] is isomorphic to a Hamming graph and conversely any Hamming graph is isomorphic to a subgraph of [Formula: see text] for some finite graded ring [Formula: see text].

Matchings in Lattice Graphs and Hamming Graphs

1994 ◽  
Vol 3 (2) ◽  
pp. 157-166
Author(s):  
Martin Aigner ◽  
Regina Klimmek
Keyword(s):  
Maximum Matching ◽  
Hamming Graph ◽  
Lattice Graph ◽  
Hamming Graphs ◽  
Dimensional Cube ◽  
Lattice Graphs

In this paper we solve the following problem on the lattice graph L(m1,…,mn) and the Hamming graph H(m1,…,mn), generalizing a result of Felzenbaum-Holzman-Kleitman on the n-dimensional cube (all mi = 2): Characterize the vectors (s1.…,sn) such that there exists a maximum matching in L, respectively, H with exactly si edges in the ith direction.

scholarly journals Structure of the space of taboo-free sequences

2020 ◽  
Vol 81 (4-5) ◽  
pp. 1029-1057
Author(s):  
Cassius Manuel ◽  
Arndt von Haeseler
Keyword(s):  
Hamming Distance ◽  
Restriction Enzymes ◽  
Finite Alphabet ◽  
Sequence Evolution ◽  
Type Ii ◽  
Hamming Graph ◽  
Recognition Sites ◽  
Restriction Sites ◽  
Vertex Set ◽  
Hamming Graphs

Abstract Models of sequence evolution typically assume that all sequences are possible. However, restriction enzymes that cut DNA at specific recognition sites provide an example where carrying a recognition site can be lethal. Motivated by this observation, we studied the set of strings over a finite alphabet with taboos, that is, with prohibited substrings. The taboo-set is referred to as $$\mathbb {T}$$ T and any allowed string as a taboo-free string. We consider the so-called Hamming graph $$\varGamma _n(\mathbb {T})$$ Γ n ( T ) , whose vertices are taboo-free strings of length n and whose edges connect two taboo-free strings if their Hamming distance equals one. Any (random) walk on this graph describes the evolution of a DNA sequence that avoids taboos. We describe the construction of the vertex set of $$\varGamma _n(\mathbb {T})$$ Γ n ( T ) . Then we state conditions under which $$\varGamma _n(\mathbb {T})$$ Γ n ( T ) and its suffix subgraphs are connected. Moreover, we provide an algorithm that determines if all these graphs are connected for an arbitrary $$\mathbb {T}$$ T . As an application of the algorithm, we show that about $$87\%$$ 87 % of bacteria listed in REBASE have a taboo-set that induces connected taboo-free Hamming graphs, because they have less than four type II restriction enzymes. On the other hand, four properly chosen taboos are enough to disconnect one suffix subgraph, and consequently connectivity of taboo-free Hamming graphs could change depending on the composition of restriction sites.

scholarly journals CONTINUITY OF UNIVERSALLY MEASURABLE HOMOMORPHISMS

2019 ◽  
Vol 7 ◽  
Author(s):  
CHRISTIAN ROSENDAL
Keyword(s):  
New York ◽  
Functional Analysis ◽  
Chromatic Number ◽  
Set Theory ◽  
Measure Theory ◽  
Polish Groups ◽  
Hamming Graph ◽  
Borel Structure ◽  
North Holland Publishing ◽  
Haar Null Sets

Answering a longstanding problem originating in Christensen’s seminal work on Haar null sets [Math. Scand. 28 (1971), 124–128; Israel J. Math. 13 (1972), 255–260; Topology and Borel Structure. Descriptive Topology and Set Theory with Applications to Functional Analysis and Measure Theory, North-Holland Mathematics Studies, 10 (Notas de Matematica, No. 51). (North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1974), iii+133 pp], we show that a universally measurable homomorphism between Polish groups is automatically continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo $\text{ZF}+\text{DC}$ , the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on $\{0,1\}^{\mathbb{N}}$ has finite chromatic number.

scholarly journals Maximal m-distance sets containing the representation of the Hamming graph H(n,m)

2017 ◽  
Vol 340 (3) ◽  
pp. 430-442 ◽  
Author(s):  
Saori Adachi ◽  
Rina Hayashi ◽  
Hiroshi Nozaki ◽  
Chika Yamamoto
Keyword(s):  
Hamming Graph ◽  
Distance Sets

scholarly journals Some Discrete Optimization Problems With Hamming and H-Comparability Graphs

2010 ◽  
Vol 27 (1-2) ◽  
pp. 167-176
Author(s):  
Tanka Nath Dhamala
Keyword(s):  
Discrete Optimization ◽  
Optimization Problems ◽  
Open Shop ◽  
Comparability Graph ◽  
Hamming Graph ◽  
Acyclic Orientation ◽  
Open Shop Problem ◽  
Sequence Graph ◽  
Hamming Graphs ◽  
Discrete Optimization Problems

Any H-comparability graph contains a Hamming graph as spanningsubgraph. An acyclic orientation of an H-comparability graph contains an acyclic orientation of the spanning Hamming graph, called sequence graph in the classical open-shop scheduling problem. We formulate different discrete optimization problems on the Hamming graphs and on H-comparability graphs and consider their complexity and relationship. Moreover, we explore the structures of these graphs in the class of irreducible sequences for the open shop problem in this paper.

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