Ars Mathematica Contemporanea (18553966)24(2)
In this paper, we first introduce a new product of finite graphs as a generalization of the X-join of graphs. We then give necessary and sufficient conditions for a graph to be isomorphic to a generalized X-join. As a main result, we give necessary and sufficient conditions under which the full automorphism group of a generalized X-join is equal to the generalized wreath product of the automorphism groups of its factors. © 2024 Society of Mathematicians, Physicists and Astronomers of Slovenia. All rights reserved.
Abdollahi, A.,
Bagherian, J.,
Ebrahimi m., M.,
Garmsiri f.m., F.M.,
Khatami bidgoli, M.,
Sobhani, R. Communications in Algebra (00927872)51(3)pp. 1011-1019
A complex character χ of a finite group G is called sharp if (Formula presented.) where (Formula presented.). In this paper we give a characterization of finite groups all non-linear irreducible characters of which are sharp. © 2022 Taylor & Francis Group, LLC.
Abdollahi, A.,
Bagherian, J.,
Jafari, F.,
Khatami bidgoli, M.,
Parvaresh, F.,
Sobhani, R. Cryptography and Communications (19362447)15(5)pp. 891-903
We give two methods that are based on the representation theory of symmetric groups to study the largest size P(n, d) of permutation codes of length n, i.e., subsets of the set Sn of all permutations on { 1 , ⋯ , n} with the minimum distance (at least) d under the Kendall τ -metric. The first method is an integer programming problem obtained from the transitive actions of Sn . The second method can be applied to refute the existence of perfect codes in Sn . Applying these methods, we reduce the known upper bound (n- 1) ! - 1 for P(n, 3) to (n-1)!-⌈n3⌉+2≤(n-1)!-2 , whenever n≥ 11 is prime. If n= 6 , 7, 11, 13, 14, 15, 17, the known upper bound for P(n, 3) is decreased by 3, 3, 9, 11, 1, 1, 4, respectively. © 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
DISCRETE MATHEMATICS (0012365X)346(1)
Let G be a finite group and Irr(G) be the set of all complex irreducible characters of G. The character-graph Delta(G) associated to G, is a graph whose vertex set is the set of primes which divide the degrees of some characters in Irr(G) and two distinct primes p and q are adjacent in Delta(G) if the product pq divides x(1), for some x is an element of Irr(G). Tong-Viet posed the conjecture that if Delta(G) is k-regular for some integer k ? 2, then Delta(G) is either a complete graph or a cocktail party graph. In this paper, we show that his conjecture is true for all regular character-graphs whose eigenvalues are in the interval [-2, infinity).(c) 2022 Elsevier B.V. All rights reserved.
Communications in Algebra (00927872)50(6)pp. 2731-2739
Let χ be a virtual (generalized) character of a finite group G and (Formula presented.) be the image of χ on (Formula presented.) The pair (Formula presented.) is said to be sharp of type L or L-sharp if (Formula presented.) If the principal character of G is not an irreducible constituent of χ, the pair (Formula presented.) is called normalized. In this paper, we first provide some counterexamples to a conjecture that was proposed by Cameron and Kiyota in 1988. This conjecture states that if (Formula presented.) is L-sharp and (Formula presented.) then the inner product (Formula presented.) is uniquely determined by L. We then prove that this conjecture is true in the case that (Formula presented.) is normalized, χ is a character of G, and L contains at least an irrational value. © 2022 Taylor & Francis Group, LLC.
Designs, Codes, and Cryptography (09251022)90(12)pp. 2841-2859
We study permutation codes which are groups and all of whose non-identity code elements have the same number of fixed points. It follows that over certain classes of groups such permutation codes exist. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
Abdollahi, A.,
Bagherian, J.,
Ebrahimi m., M.,
Garmsiri f.m., F.M.,
Khatami bidgoli, M.,
Sobhani, R. Bulletin Of The Iranian Mathematical Society (10186301)48(6)pp. 3809-3821
For a finite group G and its character χ, let Lχ be the image of χ on G- { 1 }. The pair (G, χ) is said to be sharp of type L if | G| = Π a∈L(χ(1) - a) , where L= Lχ. The pair (G, χ) is said to be normalized if the principal character of G is not an irreducible constituent of χ. In this paper, we study normalized sharp pairs of type L= { - 1 , 1 , 3 } proposed by Cameron and Kiyota in [J Algebra 115(1):125–143, 1988], under some additional hypotheses. © 2022, The Author(s) under exclusive licence to Iranian Mathematical Society.
In order to overcome the challenges posed by flash memories, the rank modulation scheme was proposed. In the rank modulation the codewords are permutations. In this paper, we study permutation codes with a specified length and minimum Kendall \tau-distance, and with as many codewords (permutations) as possible. We managed to make many significant improvements in the size of the best known codes. In particular, we show that for all n\geq 6 and for all \displaystyle \frac{3}{5}\begin{pmatrix}n\\2\end{pmatrix}\lt d\leq\frac{2}{3}\begin{pmatrix}n\\2\end{pmatrix} the largest size of a permutation code of length n and minimum distance at least d under Kendall \tau-metric is 4. © 2022 IEEE.
Abdollahi, A.,
Bagherian, J.,
Ebrahimi m., M.,
Khatami bidgoli, M.,
Shahbazi, Z.,
Sobhani, R. Czechoslovak Mathematical Journal (00114642)72(4)pp. 1081-1087
For a complex character χ of a finite group G, it is known that the product sh(χ)=∏l∈L(χ)(χ(1)−l) is a multiple of ∣G∣, where L(χ) is the image of χ on G − {1} The character χ is said to be a sharp character of type L if L = L(χ) and sh(χ) = ∣G∣. If the principal character of G is not an irreducible constituent of χ, then the character χ is called normalized. It is proposed as a problem by P. J. Cameron and M. Kiyota, to find finite groups G with normalized sharp characters of type {−1, 0, 2}. Here we prove that such a group with nontrivial center is isomorphic to the dihedral group of order 12. © 2022, Institute of Mathematics, Czech Academy of Sciences.
Discrete Mathematics (0012365X)343(11)
The generalized wreath product of permutation groups was introduced by Evdokimov and Ponomarenko in order to study the schurity problem for S-rings over cyclic groups. In this paper we construct the generalized wreath product of permutation groups by a method entirely different from Evdokimov and Ponomarenko's construction. Then we give a necessary and sufficient condition for the wedge product of schurian association schemes coming from the generalized wreath product of permutation groups. © 2020 Elsevier B.V.
Designs, Codes, and Cryptography (09251022)87(10)pp. 2335-2340
Let M(n, d) be the maximum size of a permutation code of length n and distance d. In this note, the permutation codewords of a classical code C are considered. These are the codewords with all different entries in C. Using these codewords for Reed–Solomon codes, we present some good permutation codes in this class of codes. As a consequence, since these codes are subsets of Reed–Solomon codes, decoding algorithms known for Reed–Solomon codes can also be used as a decoding algorithm for them. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.
Acta Mathematica Sinica, English Series (14398516)35(4)pp. 481-493
An irreducible character χ of an association scheme is called nonlinear if the multiplicity of χ is greater than 1. The main result of this paper gives a characterization of commutative association schemes with at most two nonlinear irreducible characters. This yields a characterization of finite groups with at most two nonlinear irreducible characters. A class of noncommutative association schemes with at most two nonlinear irreducible character is also given. © 2018, Springer-Verlag GmbH Germany & The Editorial Office of AMS.
Bulletin of the Iranian Mathematical Society (10186301)45(5)pp. 1515-1529
In this paper, we first show that the group scheme of a Camina triple has the wedge product structure of association schemes. Then as a main result, we give a characterization of Camina triples in terms of their irreducible characters. © 2019, Iranian Mathematical Society.
Journal of Algebra and its Applications (02194988)18(4)
In this paper, we first show that the wedge product of a thin association scheme and a schurian association scheme is schurian. Then as an application of this result, we investigate the schurity problem for the association schemes having the thin radical series. We show that these association schemes are schurian under some conditions on the successive quotients of their thin radical series. © 2019 World Scientific Publishing Company.
Communications in Algebra (00927872)46(5)pp. 2179-2193
The main result of this paper gives a classification of commutative association schemes, all irreducible characters of which have multiplicity 1 or a prime p. © 2017 Taylor & Francis.
Communications in Algebra (00927872)44(9)pp. 3692-3704
From Burnside's pαqβ-Theorem, it follows that any nonabelian group of order pαqβ, where p and q are primes, cannot be simple. As a main result of this article, we state and prove an analog of the mentioned theorem for commutative association schemes. © 2016, Copyright © Taylor & Francis Group, LLC.
Canadian Mathematical Bulletin (00084395)57(2)pp. 231-239
In this paper we show that every module of a table algebra can be considered as a faithful module of some quotient table algebra. Also we prove that every faithful module of a table algebra determines a closed subset that is a cyclic group. As a main result we give some information about multiplicities of characters in table algebras. © Canadian Mathematical Society 2013.
Discrete Mathematics (0012365X)313(10)pp. 1112-1118
In this paper we give some properties of the character values of nilpotent commutative association schemes. As a main result, a class of commutative Schurian association schemes is given. © 2013 Elsevier B.V. All rights reserved.
Electronic Journal of Combinatorics (10778926)18(1)pp. 1-10
In the character theory of finite groups the Burnside-Brauer Theorem is a well- known result which deals with products of characters in finite groups. In this paper, we first define the character products for table algebras and then by observing the relationship between the characters of a table algebra and the characters of its quotient, we provide a condition in which the products of characters of table algebras are characters. As a main result we state and prove the Burnside-Brauer Theorem on finite groups for table algebras.
Electronic Journal of Combinatorics (10778926)17(1)pp. 1-13
It is well known that the complex adjacency algebra A of an association scheme has a specific module, namely the standard module, that contains the regular module of A as a submodule. The character afforded by the standard module is called the standard character. In this paper we first define the concept of standard character for C-algebras and we say that a C-algebra has the standard character condition if it admits the standard character. Among other results we acquire a necessary and sufficient condition for a table algebra to originate from an association scheme. Finally, we prove that given a C-algebra admits the standard character and its all degrees are integers if and only if so its dual.
Journal of Algebraic Combinatorics (09259899)27(2)pp. 173-185
We introduce a concept of cyclotomic association scheme over a finite near-field K. It is proved that any isomorphism of two such nontrivial schemes is induced by a suitable element of the group AGL(V), where V is the linear space associated with K. A sufficient condition on a cyclotomic scheme C that guarantee the inclusion Aut(C)≤ ≤{A} Γ L(1,F), where F is a finite field with |K| elements, is given. © 2007 Springer Science+Business Media, LLC.
