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.

scholarly journals Separability Number and Schurity Number of Coherent Configurations

10.37236/1509 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Sergei Evdokimov ◽  
Ilia Ponomarenko
Keyword(s):  
Automorphism Group ◽  
Regular Graph ◽  
Cartesian Product ◽  
Distance Regular Graph ◽  
Coherent Configuration ◽  
Hamming Graph ◽  
Point Set ◽  
Configuration Scheme ◽  
Positive Integers ◽  
Cyclotomic Scheme

To each coherent configuration (scheme) ${\cal C}$ and positive integer $m$ we associate a natural scheme $\widehat{\cal C}^{(m)}$ on the $m$-fold Cartesian product of the point set of ${\cal C}$ having the same automorphism group as ${\cal C}$. Using this construction we define and study two positive integers: the separability number $s({\cal C})$ and the Schurity number $t({\cal C})$ of ${\cal C}$. It turns out that $s({\cal C})\le m$ iff ${\cal C}$ is uniquely determined up to isomorphism by the intersection numbers of the scheme $\widehat{\cal C}^{(m)}$. Similarly, $t({\cal C})\le m$ iff the diagonal subscheme of $\widehat{\cal C}^{(m)}$ is an orbital one. In particular, if ${\cal C}$ is the scheme of a distance-regular graph $\Gamma$, then $s({\cal C})=1$ iff $\Gamma$ is uniquely determined by its parameters whereas $t({\cal C})=1$ iff $\Gamma$ is distance-transitive. We show that if ${\cal C}$ is a Johnson, Hamming or Grassmann scheme, then $s({\cal C})\le 2$ and $t({\cal C})=1$. Moreover, we find the exact values of $s({\cal C})$ and $t({\cal C})$ for the scheme ${\cal C}$ associated with any distance-regular graph having the same parameters as some Johnson or Hamming graph. In particular, $s({\cal C})=t({\cal C})=2$ if ${\cal C}$ is the scheme of a Doob graph. In addition, we prove that $s({\cal C})\le 2$ and $t({\cal C})\le 2$ for any imprimitive 3/2-homogeneous scheme. Finally, we show that $s({\cal C})\le 4$, whenever ${\cal C}$ is a cyclotomic scheme on a prime number of points.

scholarly journals Fundamental Homomorphism Theorems for Neutrosophic Extended Triplet Groups

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 321 ◽  
Author(s):  
Mehmet Çelik ◽  
Moges Shalla ◽  
Necati Olgun
Keyword(s):  
Group Theory ◽  
Significant Role ◽  
Classical Group ◽  
Algebraic Structures ◽  
Algebraic Systems ◽  
Homomorphism Theorem ◽  
Special Cases ◽  
Isomorphism Theorems

In classical group theory, homomorphism and isomorphism are significant to study the relation between two algebraic systems. Through this article, we propose neutro-homomorphism and neutro-isomorphism for the neutrosophic extended triplet group (NETG) which plays a significant role in the theory of neutrosophic triplet algebraic structures. Then, we define neutro-monomorphism, neutro-epimorphism, and neutro-automorphism. We give and prove some theorems related to these structures. Furthermore, the Fundamental homomorphism theorem for the NETG is given and some special cases are discussed. First and second neutro-isomorphism theorems are stated. Finally, by applying homomorphism theorems to neutrosophic extended triplet algebraic structures, we have examined how closely different systems are related.

scholarly journals Yet Another Distance Regular Graph Related to a Golay Code

10.37236/1220 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Leonard H. Soicher
Keyword(s):  
Automorphism Group ◽  
Regular Graph ◽  
Intersection Array ◽  
Distance Regular Graph ◽  
Golay Code ◽  
Transitive Graph

We describe a new distance-regular, but not distance-transitive, graph. This graph has intersection array $\{110,81,12;1,18,90\}$, and automorphism group $M_{22}\colon 2$.

A computer proof of relations in a certain class of groups

1991 ◽  
Vol 117 (1-2) ◽  
pp. 109-114
Author(s):  
Edmund F. Robertson ◽  
Kevin Rutherford
Keyword(s):  
Group Theory ◽  
Special Cases ◽  
Computer Proof

SynopsisA gp-toolkit consisting of computer implementations of various group theory methods, in particular a Tietze transformation program, was designed. Special cases of a conjecture were solved by the gp-toolkit. Examination of the method used by the gp-toolkit to deduce relations showed that a general approach had been employed. We present a proof verifying that the conjecture is true which is a straightforward generalisation of the method discovered by the gp-toolkit.

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 Introducing edge-biregular maps

2021 ◽  
Author(s):  
Olivia Reade
Keyword(s):  
Automorphism Group ◽  
Euler Characteristic ◽  
Dihedral Group ◽  
Automorphism Groups ◽  
Triangle Group ◽  
Supporting Surface ◽  
Regular Maps ◽  
Special Cases ◽  
Edge Colourings

AbstractWe introduce the concept of alternate-edge-colourings for maps and study highly symmetric examples of such maps. Edge-biregular maps of type (k, l) occur as smooth normal quotients of a particular index two subgroup of $$T_{k,l}$$ T k , l , the full triangle group describing regular plane (k, l)-tessellations. The resulting colour-preserving automorphism groups can be generated by four involutions. We explore special cases when the usual four generators are not distinct involutions, with constructions relating these maps to fully regular maps. We classify edge-biregular maps when the supporting surface has non-negative Euler characteristic, and edge-biregular maps on arbitrary surfaces when the colour-preserving automorphism group is isomorphic to a dihedral group.

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.

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