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Journal articles on the topic 'Cayley permutations'

Author: Grafiati

Published: 10 March 2023

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1

CHITTURI, BHADRACHALAM. "UPPER BOUNDS FOR SORTING PERMUTATIONS WITH A TRANSPOSITION TREE." Discrete Mathematics, Algorithms and Applications 05, no. 01 (March 2013): 1350003. http://dx.doi.org/10.1142/s1793830913500031.

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An upper bound for sorting permutations with an operation estimates the diameter of the corresponding Cayley graph and an exact upper bound equals the diameter. Computing tight upper bounds for various operations is of theoretical and practical (e.g., interconnection networks, genetics) interest. Akers and Krishnamurthy gave a Ω(n! n2) time method that examines n! permutations to compute an upper bound, f(Γ), to sort any permutation with a given transposition tree T, where Γ is the Cayley graph corresponding to T. We compute two intuitive upper bounds γ and δ′ each in O(n2) time for the same, by working solely with the transposition tree. Recently, Ganesan computed β, an estimate of the exact upper bound for the same, in O(n2) time. Our upper bounds are tighter than f(Γ) and β, on average and in most of the cases. For a class of trees, we prove that the new upper bounds are tighter than β and f(Γ).

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2

Olshevskyi, M. S. "Metric properties of Cayley graphs of alternating groups." Carpathian Mathematical Publications 13, no. 2 (November 19, 2021): 545–81. http://dx.doi.org/10.15330/cmp.13.2.545-581.

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A well known diameter search problem for finite groups with respect to its systems of generators is considered. The problem can be formulated as follows: find the diameter of a group over its system of generators. The diameter of a group over a specific system of generators is the diameter of the corresponding Cayley graph. It is considered alternating groups with classic irreducible system of generators consisting of cycles with length three of the form $(1,2,k)$. The main part of the paper concentrates on analysis how even permutations decompose with respect to this system of generators. The rules for moving generators from permutation's decomposition from left to right and from right to left are introduced. These rules give rise for transformations of decompositions, that do not increase their lengths. They are applied for removing fixed points of a permutation, that were included in its decomposition. Based on this rule the stability of system of generators is proved. The strict growing property of the system of generators is also proved, as the corollary of transformation rules and the stability property. It is considered homogeneous theory, that was introduced in the previous author's paper. For the series of alternating groups with systems of generators mentioned above it is shown that this series is uniform and homogeneous. It makes possible to apply the homogeneous down search algorithm to compute the diameter. This algorithm is applied and exact values of diameters for alternating groups of degree up to 43 are computed.

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3

Olshevskyi, M. "The lower bound of diameter of Alternating groups." Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 4 (2021): 11–22. http://dx.doi.org/10.17721/1812-5409.2021/4.1.

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In this paper we consider a specific case of the diameter search problem for finite groups, thecase where the system of generators is fixed. This problem is well-known and can be formulated in the following way: find the diameter of a group over its system of generators. The diameter of the corresponding Cayley graph is the diameter of a group over its specific system of generators. The main object of the research is the alternating group with the system of generators consisting of cycles having length three and the form (1,2,k). This system of generators is a classical irreducible system of generators of the alternating group. It is introduced the property of even permutations to be balanced. We consider the set of balanced permutations and permutations close enough to balanced and find minimum decompositions of them over defined system of generators. The main result of the paper is the lower bound of the diameter of Alternating group over con-sidered system of generators. The estimation is achieved using minimal decompositions of balanced permutations.

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4

Babai, L., and G. L. Hetyei. "On the Diameter of Random Cayley Graphs of the Symmetric Group." Combinatorics, Probability and Computing 1, no. 3 (September 1992): 201–8. http://dx.doi.org/10.1017/s0963548300000237.

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Let σ, π be two permutations selected at random from the uniform distribution on the symmetric group Sn. By a result of Dixon [5], the subgroup G generated by σ, π is almost always (i.e. with probability approaching 1 as n → ∞) either Sn or the alternating group An. We prove that the diameter of the Cayley graph of G defined by {σ, π} is almost always not greater than exp ((½ + o(l)). (In n)2).

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5

Abdesselam, B., and A. Chakrabarti. "Multiparameter Statistical Models from Braid Matrices: Explicit Eigenvalues of Transfer Matrices , Spin Chains, Factorizable Scatterings for All." Advances in Mathematical Physics 2012 (2012): 1–21. http://dx.doi.org/10.1155/2012/193190.

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For a class of multiparameter statistical models based on braid matrices, the eigenvalues of the transfer matrix are obtained explicitly for all . Our formalism yields them as solutions of sets of linear equations with simple constant coefficients. The role of zero-sum multiplets constituted in terms of roots of unity is pointed out, and their origin is traced to circular permutations of the indices in the tensor products of basis states induced by our class of matrices. The role of free parameters, increasing as withN, is emphasized throughout. Spin chain Hamiltonians are constructed and studied for allN. Inverse Cayley transforms of the Yang-Baxter matrices corresponding to our braid matrices are obtained for allN. They provide potentials for factorizableS-matrices. Main results are summarized, and perspectives are indicated in the concluding remarks.

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6

Păun, Udrea. "$G$ method in action: Fast exact sampling from set of permutations of order $n$ according to Mallows model through Cayley metric." Brazilian Journal of Probability and Statistics 31, no. 2 (May 2017): 338–52. http://dx.doi.org/10.1214/16-bjps316.

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7

Skresanov, Saveliy V. "Subgroups of minimal index in polynomial time." Journal of Algebra and Its Applications 19, no. 01 (January 29, 2019): 2050010. http://dx.doi.org/10.1142/s0219498820500103.

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By applying an old result of Y. Berkovich, we provide a polynomial-time algorithm for computing the minimal possible index of a proper subgroup of a finite permutation group [Formula: see text]. Moreover, we find that subgroup explicitly and within the same time if [Formula: see text] is given by a Cayley table. As a corollary, we get an algorithm for testing whether or not a finite permutation group acts on a tree non-trivially.

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8

Alspach, Brian, and Shaofei Du. "Suborbit Structure of Permutation p-Groups and an Application to Cayley Digraph Isomorphism." Canadian Mathematical Bulletin 47, no. 2 (June 1, 2004): 161–67. http://dx.doi.org/10.4153/cmb-2004-017-9.

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AbstractLet P be a transitive permutation group of order pm, p an odd prime, containing a regular cyclic subgroup. The main result of this paper is a determination of the suborbits of P. The main result is used to give a simple proof of a recent result by J. Morris on Cayley digraph isomorphisms.

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9

LI, CAI HENG, and CHERYL E. PRAEGER. "SELF-COMPLEMENTARY VERTEX-TRANSITIVE GRAPHS NEED NOT BE CAYLEY GRAPHS." Bulletin of the London Mathematical Society 33, no. 6 (November 2001): 653–61. http://dx.doi.org/10.1112/s0024609301008505.

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A construction is given of an infinite family of finite self-complementary, vertex-transitive graphs which are not Cayley graphs. To the authors' knowledge, these are the first known examples of such graphs. The nature of the construction was suggested by a general study of the structure of self-complementary, vertex-transitive graphs. It involves the product action of a wreath product of permutation groups.

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10

Kuznetsov, А. A., and V. V. Kishkan. "A ROUTING ALGORITHM FOR THE CAYLEY GRAPHS GENERATED BY PERMUTATION GROUPS." Siberian Journal of Science and Technology 21, no. 2 (2020): 187–94. http://dx.doi.org/10.31772/2587-6066-2020-21-2-187-194.

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11

Dalfó, C., and M. A. Fiol. "Spectra and eigenspaces from regular partitions of Cayley (di)graphs of permutation groups." Linear Algebra and its Applications 597 (July 2020): 94–112. http://dx.doi.org/10.1016/j.laa.2020.03.015.

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12

Lakshmivarahan, S., Jung-Sing Jwo, and S. K. Dhall. "Symmetry in interconnection networks based on Cayley graphs of permutation groups: A survey." Parallel Computing 19, no. 4 (April 1993): 361–407. http://dx.doi.org/10.1016/0167-8191(93)90054-o.

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13

Wei, D. S. L., F. P. Muga, and K. Naik. "Isomorphism of degree four Cayley graph and wrapped butterfly and their optimal permutation routing algorithm." IEEE Transactions on Parallel and Distributed Systems 10, no. 12 (1999): 1290–98. http://dx.doi.org/10.1109/71.819950.

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14

Temme, F. P. "Cayleyan ?n-encoded SU(2)�?n?? embeddings: Nuclear spin permutation symmetries via polyhedral lattice-point models, for modulo-i ?(?i(?n??)) combinatorial invariance sets." International Journal of Quantum Chemistry 78, no. 2 (2000): 71–82. http://dx.doi.org/10.1002/(sici)1097-461x(2000)78:2<71::aid-qua1>3.0.co;2-t.

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15

Temme, F. P. "Mathematical determinacy properties of specific automorphic FG/ group embeddings of NMR spin permutational (CNP) algebras, beyond strictly Cayleyan models: a new role for “induced symmetry”." Journal of Molecular Structure: THEOCHEM 578, no. 1-3 (February 2002): 145–57. http://dx.doi.org/10.1016/s0166-1280(01)00697-2.

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16

Huang, Zejun, Chi-Kwong Li, Sharon H. Li, and Nung-Sing Sze. "Factorization of Permutations." Electronic Journal of Linear Algebra 30 (February 8, 2015). http://dx.doi.org/10.13001/1081-3810.2849.

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The problem of factoring a permutation as a product of special types of transpositions, namely, those transpositions involving two positions with bounded distances, is considered. In particular, the minimum number, Î´, such that every permutation can be factored into no more than Î´ special transpositions is investigated. This study is related to sorting algorithms, Cayley graphs, and genomics.

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17

Brunat, Josep M. "Explicit Cayley Covers of Kautz Digraphs." Electronic Journal of Combinatorics 18, no. 1 (May 8, 2011). http://dx.doi.org/10.37236/592.

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Given a finite set $V$ and a set $S$ of permutations of $V$, the group action graph $\mathrm{GAG}(V,S)$ is the digraph with vertex set $V$ and arcs $(v,v^\sigma)$ for all $v\in V$ and $\sigma\in S$. Let $\langle S\rangle$ be the group generated by $S$. The Cayley digraph $\textrm{Cay}(\langle S\rangle, S)$ is called a Cayley cover of $\mathrm{GAG}(V,S)$. We define the Kautz digraphs as group action graphs and give an explicit construction of the corresponding Cayley cover. This is an answer to a problem posed by Heydemann in 1996.

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18

Liu, Xiaoqing, Shuming Zhou, and Hong Zhang. "Cyclic Vertex (Edge) Connectivity of Burnt Pancake Graphs." Parallel Processing Letters, September 6, 2022. http://dx.doi.org/10.1142/s0129626422500062.

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The cyclic vertex (resp., edge) connectivity of a graph [Formula: see text], denoted by [Formula: see text] (resp., [Formula: see text]), is the minimum number of vertices (resp., edges) whose removal from [Formula: see text] results in a disconnected graph and at least two remaining components contain cycles. Thus, to determine the exact values of [Formula: see text] and [Formula: see text] is important in the reliability assessment of interconnection networks. However, the study of the cyclic vertex (edge) connectivity is less involved. In this paper, we determine the cyclic vertex (edge) connectivity of the burnt pancake graphs [Formula: see text] which is the Cayley graph of the group of signed permutations using prefix reversals as generators. By exploring the combinatorial properties and fault-tolerance of [Formula: see text], we show [Formula: see text] and [Formula: see text] for [Formula: see text]. Moreover, we determine that [Formula: see text] for [Formula: see text].

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19

Curtin, Eugene. "Cubic Cayley graphs with small diameter." Discrete Mathematics & Theoretical Computer Science Vol. 4 no. 2 (January 1, 2001). http://dx.doi.org/10.46298/dmtcs.285.

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International audience In this paper we apply Polya's Theorem to the problem of enumerating Cayley graphs on permutation groups up to isomorphisms induced by conjugacy in the symmetric group. We report the results of a search of all three-regular Cayley graphs on permutation groups of degree at most nine for small diameter graphs. We explore several methods of constructing covering graphs of these Cayley graphs. Examples of large graphs with small diameter are obtained.

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20

LI, CAI HENG, GUANG RAO, and SHU JIAO SONG. "NEW CONSTRUCTIONS OF SELF-COMPLEMENTARY CAYLEY GRAPHS." Journal of the Australian Mathematical Society, January 12, 2021, 1–14. http://dx.doi.org/10.1017/s1446788720000488.

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Abstract Vertex-primitive self-complementary graphs were proved to be affine or in product action by Guralnick et al. [‘On orbital partitions and exceptionality of primitive permutation groups’, Trans. Amer. Math. Soc.356 (2004), 4857–4872]. The product action type is known in some sense. In this paper, we provide a generic construction for the affine case and several families of new self-complementary Cayley graphs are constructed.

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21

Yee Siang, Gan, Fong Wan Heng, Nor Haniza Sarmin, and Sherzod Turaev. "Permutation Groups in Automata Diagrams." Malaysian Journal of Fundamental and Applied Sciences 9, no. 1 (January 25, 2013). http://dx.doi.org/10.11113/mjfas.v9n1.79.

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Automata act as classical models for recognition devices. From the previous researches, the classical models of automata have been used to scan strings and to determine the types of languages a string belongs to. In the study of automata and group theory, it has been found that the properties of a group can be recognized by the automata using the automata diagrams. There are two types of automata used to study the properties of a group, namely modified finite automata and modified Watson-Crick finite automata. Thus, in this paper, automata diagrams are constructed to recognize permutation groups using the data given by the Cayley table. Thus, the properties of permutation group are analyzed using the automaton diagram that has been constructed. Moreover, some theorems for the properties of permutation group in term of automata are also given in this paper.

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22

Bergeron-Brlek, Anouk. "Words and Noncommutative Invariants of the Hyperoctahedral Group." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AN,..., Proceedings (January 1, 2010). http://dx.doi.org/10.46298/dmtcs.2870.

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International audience Let $\mathcal{B}_n$ be the hyperoctahedral group acting on a complex vector space $\mathcal{V}$. We present a combinatorial method to decompose the tensor algebra $T(\mathcal{V})$ on $\mathcal{V}$ into simple modules via certain words in a particular Cayley graph of $\mathcal{B}_n$. We then give combinatorial interpretations for the graded dimension and the number of free generators of the subalgebra $T(\mathcal{V})^{\mathcal{B}_n}$ of invariants of $\mathcal{B}_n$, in terms of these words, and make explicit the case of the signed permutation module. To this end, we require a morphism from the Mantaci-Reutenauer algebra onto the algebra of characters due to Bonnafé and Hohlweg. Soit $\mathcal{B}_n$ le groupe hyperoctaédral agissant sur un espace vectoriel complexe $\mathcal{V}$. Nous présentons une méthode combinatoire donnant la décomposition de l'algèbre $T(\mathcal{V})$ des tenseurs sur $\mathcal{V}$ en modules simples via certains mots dans un graphe de Cayley donné. Nous donnons ensuite des interprétations combinatoires pour la dimension graduée et le nombre de générateurs libres de la sous-algèbre $T(\mathcal{V})^{\mathcal{B}_n}$ des invariants de $\mathcal{B}_n$, en termes de ces mots, et explicitons le cas du module de permutation signé. À cette fin, nous utilisons un morphisme entre l'algèbre de Mantaci-Reutenauer et l'algèbre des caractères introduit par Bonnafé et Hohlweg.

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23

Mitamura, Takuma, and Akira Tanaka. "Minimum upper bound of the Cayley transform of an orthogonal matrix multiplied by signed permutation matrices." Linear Algebra and its Applications, November 2022. http://dx.doi.org/10.1016/j.laa.2022.11.009.

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24

Watkins, Mark E. "Infinite Graphical Frobenius Representations." Electronic Journal of Combinatorics 25, no. 4 (October 19, 2018). http://dx.doi.org/10.37236/7294.

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A graphical Frobenius representation (GFR) of a Frobenius (permutation) group $G$ is a graph $\Gamma$ whose automorphism group Aut$(\Gamma)$ acts as a Frobenius permutation group on the vertex set of $\Gamma$, that is, Aut$(\Gamma)$ acts vertex-transitively with the property that all nonidentity automorphisms fix either exactly one or zero vertices and there are some of each kind. The set $K$ of all fixed-point-free automorphisms together with the identity is called the kernel of $G$. Whenever $G$ is finite, $K$ is a regular normal subgroup of $G$ (F. G. Frobenius, 1901), in which case $\Gamma$ is a Cayley graph of $K$. The same holds true for all the infinite instances presented here.Infinite, locally finite, vertex-transitive graphs can be classified with respect to (i) the cardinality of their set of ends and (ii) their growth rate. We construct families of infinite GFRs for all possible combinations of these two properties. There exist infinitely many GFRs with polynomial growth of degree $d$ for every positive integer $d$, and there exist infinite families of GFRs of exponential growth, both $1$-ended and infinitely-ended, that underlie infinite chiral planar maps. There also exist GFRs of free products of finitely many finitely generated groups. Graphs of connectivity 1 having a Frobenius automorphism group are characterized.

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