An infinite family of cubic one-regular graphs with unsolvable automorphism groups

Abstract We study the automorphisms of a function field of genus g ≥ 2 over an algebraically closed field of characteristic p > 0. More precisely, we show that the order of a nilpotent subgroup G of its automorphism group is bounded by 16 (g – 1) when G is not a p-group. We show that if |G| = 16(g – 1), then g – 1 is a power of 2. Furthermore, we provide an infinite family of function fields attaining the bound.

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An infinite family of braid group representations in automorphism groups of free groups

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216520420079 ◽

2020 ◽

Vol 29 (10) ◽

pp. 2042007

Author(s):

Wonjun Chang ◽

Byung Chun Kim ◽

Yongjin Song

Keyword(s):

Braid Group ◽

Class Group ◽

Automorphism Groups ◽

Free Groups ◽

Infinite Family ◽

Braid Groups ◽

Mapping Class ◽

Group Representations ◽

Class Groups ◽

Branched Coverings

The [Formula: see text]-fold ([Formula: see text]) branched coverings on a disk give an infinite family of nongeometric embeddings of braid groups into mapping class groups. We, in this paper, give new explicit expressions of these braid group representations into automorphism groups of free groups in terms of the actions on the generators of free groups. We also give a systematic way of constructing and expressing these braid group representations in terms of a new gadget, called covering groupoid. We prove that each generator [Formula: see text] of braid group inside mapping class group induced by [Formula: see text]-fold covering is the product of [Formula: see text] Dehn twists on the surface.

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Codes from incidence matrices of some regular graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s179383092150035x ◽

2020 ◽

pp. 2150035

Author(s):

R. Saranya ◽

C. Durairajan

Keyword(s):

Linear Codes ◽

Automorphism Groups ◽

Regular Graphs ◽

Permutation Decoding ◽

Incidence Matrices ◽

Hamming Metric ◽

Vertex Set ◽

Lee Distance ◽

Lee Metric ◽

Transitive Subgroup

We examine the [Formula: see text]-ary linear codes with respect to Lee metric from incidence matrix of the Lee graph with vertex set [Formula: see text] and two vertices being adjacent if their Lee distance is one. All the main parameters of the codes are obtained as [Formula: see text] if [Formula: see text] is odd and [Formula: see text] if [Formula: see text] is even. We examine also the [Formula: see text]-ary linear codes with respect to Hamming metric from incidence matrices of Desargues graph, Pappus graph, Folkman graph and the main parameters of the codes are [Formula: see text], respectively. Any transitive subgroup of automorphism groups of these graphs can be used for full permutation decoding using the corresponding codes. All the above codes can be used for full error correction by permutation decoding.

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On the automorphism groups of strongly regular graphs I

Proceedings of the 5th conference on Innovations in theoretical computer science - ITCS '14 ◽

10.1145/2554797.2554830 ◽

2014 ◽

Cited By ~ 2

Author(s):

László Babai

Keyword(s):

Automorphism Groups ◽

Strongly Regular Graphs ◽

Regular Graphs ◽

Strongly Regular

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On Arc-Transitive Metacyclic Covers of Graphs with Order Twice a Prime

The Electronic Journal of Combinatorics ◽

10.37236/7146 ◽

2018 ◽

Vol 25 (3) ◽

Author(s):

Zhaohong Huang ◽

Jiangmin Pan

Keyword(s):

Bipartite Graph ◽

Prime Order ◽

Complete Bipartite Graph ◽

Infinite Family ◽

Petersen Graph ◽

Regular Graphs ◽

Dihedral Groups ◽

Hadamard Design ◽

Symmetric Graphs ◽

Abelian Covers

Quite a lot of attention has been paid recently to the characterization and construction of edge- or arc-transitive abelian (mostly cyclic or elementary abelian) covers of symmetric graphs, but there are rare results for nonabelian covers since the voltage graph techniques are generally not easy to be used in this case. In this paper, we will classify certain metacyclic arc-transitive covers of all non-complete symmetric graphs with prime valency and twice a prime order $2p$ (involving the complete bipartite graph ${\sf K}_{p,p}$, the Petersen graph, the Heawood graph, the Hadamard design on $22$ points and an infinite family of prime-valent arc-regular graphs of dihedral groups). A few previous results are extended.

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Switching of Edges in Strongly Regular Graphs I: A Family of Partial Difference Sets on 100 Vertices

The Electronic Journal of Combinatorics ◽

10.37236/1710 ◽

2003 ◽

Vol 10 (1) ◽

Cited By ~ 14

Author(s):

L. K. Jørgensen ◽

M. Klin

Keyword(s):

Automorphism Groups ◽

Difference Sets ◽

Association Schemes ◽

Strongly Regular Graphs ◽

Regular Graphs ◽

Partial Difference ◽

Galois Correspondence ◽

Partial Difference Sets ◽

Strongly Regular ◽

Open Question

We present 15 new partial difference sets over 4 non-abelian groups of order 100 and 2 new strongly regular graphs with intransitive automorphism groups. The strongly regular graphs and corresponding partial difference sets have the following parameters: (100,22,0,6), (100,36,14,12), (100,45,20,20), (100,44,18,20). The existence of strongly regular graphs with the latter set of parameters was an open question. Our method is based on combination of Galois correspondence between permutation groups and association schemes, classical Seidel's switching of edges and essential use of computer algebra packages. As a by-product, a few new amorphic association schemes with 3 classes on 100 points are discovered.

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A Construction of Small $(q-1)$-Regular Graphs of Girth 8

The Electronic Journal of Combinatorics ◽

10.37236/4397 ◽

2015 ◽

Vol 22 (2) ◽

Cited By ~ 1

Author(s):

M. Abreu ◽

G. Araujo-Pardo ◽

C. Balbuena ◽

D. Labbate

Keyword(s):

Prime Power ◽

Infinite Family ◽

Regular Graphs

In this note we construct a new infinite family of $(q-1)$-regular graphs of girth 8 and order $2q(q-1)^2$ for all prime powers $q\geq 16$, which are the smallest known so far whenever $q-1$ is not a prime power or a prime power plus one itself.

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Rectilinear Monotone r-Regular Planar Graphs for r = {3, 4, 5}

d'CARTESIAN ◽

10.35799/dc.4.1.2015.8318 ◽

2015 ◽

Vol 4 (1) ◽

pp. 103

Author(s):

Arthur Wulur ◽

Benny Pinontoan ◽

Mans Mananohas

Keyword(s):

Planar Graph ◽

Connected Graph ◽

Planar Graphs ◽

Infinite Family ◽

Regular Graphs ◽

Vertex Set ◽

Simple Connected Graph

A graph G consists of non-empty set of vertex/vertices (also called node/nodes) and the set of lines connecting two vertices called edge/edges. The vertex set of a graph G is denoted by V(G) and the edge set is denoted by E(G). A Rectilinear Monotone r-Regular Planar Graph is a simple connected graph that consists of vertices with same degree and horizontal or diagonal straight edges without vertical edges and edges crossing. This research shows that there are infinite family of rectilinear monotone r-regular planar graphs for r = 3and r = 4. For r = 5, there are two drawings of rectilinear monotone r-regular planar graphs with 12 vertices and 16 vertices. Keywords: Monotone Drawings, Planar Graphs, Rectilinear Graphs, Regular Graphs

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Automorphism Groups of Jordan Algebras

Nagoya Mathematical Journal ◽

10.1017/s0027763000026647 ◽

1968 ◽

Vol 32 ◽

pp. 227-235

Author(s):

E. Ray Bobo

Keyword(s):

Vector Space ◽

Automorphism Groups ◽

Infinite Family ◽

Jordan Algebras ◽

Structure Theory ◽

Important Class ◽

Natural Basis ◽

Generators And Relations ◽

Characteristic Two ◽

Central Simple

In his development of a structure theory for Jordan algebras of characteristic two, E.C. Paige [1] introduces an important class of central simple Jordan algebras S[2n], It is the purpose of this paper to completely determine the automorphism groups of the algebras S[2n]. The automorphisms will be represented as matrices operating on a natural basis for the underlying vector space of the algebra. Using this characterization, generators and relations will be obtained for each of the automorphism groups. In this way, we will produce an infinite family of finite 2-groups.

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