arXiv subjects

Roghayeh Hafezieh

Publications and source records attributed to Roghayeh Hafezieh.

7 recordsLinked to original sources

arXiv2026

On the permutation automorphisms of binary cubic codes

A binary linear code whose permutation automorphism group has a fixed point free permutation of order $3$ is called a binary cubic code. The scope of this paper is to investigate the structural properties of binary cubic codes. Let $C$ be a binary cubic $[n,k]$ code. In this paper, we prove that if $n\geq 30$ and $C$ has permutation automorphism group of order three, then $k\geq 6$. Additionally, we show that if $n < 30$ and $k\leq 4$, then the permutation automorphism group of $C$ has order greater than three. Moreover, along the way, we provide some results on the structure of the higher dimensional cubic codes. In particular, we present some results concerning the structure of the putative extremal self-dual $[72,36,16]$ code under the assumption that it is cubic.

cs.IT↗

arXiv2024

On the structure of the character degree graphs having diameter three

The structure of the character degree graphs $Δ(G)$, i.e. the prime graphs on the set $\mathrm{cd}(G)$ of the irreducible character degrees of a finite group $G$, such that $G$ is solvable and $Δ(G)$ has diameter three, remains an intriguing area of study. However, a comprehensive understanding of these structures remains elusive. In this paper, we prove some properties and provide an infinite series of examples of this class of graphs, building on the ideas of Mark Lewis.

math.GR↗

arXiv2022

Involutory permutation automorphisms of binary linear codes

We investigate the properties of binary linear codes of even length whose permutation automorphism group is a cyclic group generated by an involution. Up to dimension or co-dimension $4$, we show that there is no quasi group code whose permutation automorphism group is isomorphic to $C_2$. By generalizing the method we use to prove this result, we obtain results on the structure of putative extremal self-dual $[72, 36, 16]$ and $[96, 48, 20]$ codes in the presence of an involutory permutation automorphism.

cs.IT↗

arXiv2019

The influence of cut vertices and eigenvalues on character graphs of solvable groups

Given a finite group $G$, the character graph, denoted by $Δ(G)$, for its irreducible character degrees is a graph with vertex set $ρ(G)$ which is the set of prime numbers that divide the irreducible character degrees of $G$, and with $\{p,q\}$ being an edge if there exist a non-linear $χ\in {\rm Irr}(G)$ whose degree is divisible by $pq$. In this paper, we discuss the influences of cut vertices and eigenvalues of $Δ(G)$ on the group structure of $G$. Recently, Lewis and Meng proved the character graph of each solvable group has at most one cut vertex. Now, we determine the structure of character graphs of solvable groups with a cut vertex and diameter $3$. Furthermore, we study solvable groups whose character graphs have at most two distinct eigenvalues. Moreover, we investigate the solvable groups whose character graphs are regular with three distinct eigenvalues. In addition, we give some lower bounds for the number of edges of $Δ(G)$.

math.GR↗

arXiv2019

An overview on the bipartite divisor graph for the set of irreducible character degrees

Let $G$ be a finite group. The bipartite divisor graph for the set of irreducible complex character degrees is the undirected graph with vertex set consisting of the prime numbers dividing some character degree and of the non-identity character degrees, where a prime number $p$ is declared to be adjacent to a character degree $m$ if and only if $p$ divides $m$. This graph is bipartite and it encodes two of the most widely studied graphs associated to the character degrees of a finite group: the prime graph and the divisor graph on the set of irreducible character degrees. The scope of this paper is two-fold. We draw some attention to the bipartite divisor graph for the set of irreducible complex character degrees by outlining the main results that have been proved so far. In this process we improve some of these results and we leave some open problems.

math.GR↗

arXiv2015

Bipartite divisor graph for the set of irreducible character degrees

Let $G$ be a finite group. We consider the set of the irreducible complex characters of $G$, namely $Irr(G)$, and the related degree set $cd(G)=\{χ(1) : χ\in Irr(G)\}$. Let $ρ(G)$ be the set of all primes which divide some character degree of $G$. In this paper we introduce the bipartite divisor graph for $cd(G)$ as an undirected bipartite graph with vertex set $ρ(G)\cup (cd(G)\setminus\{1\})$, such that an element $p$ of $ρ(G)$ is adjacent to an element $m$ of $cd(G)\setminus\{1\}$ if and only if $p$ divides $m$. We denote this graph simply by $B(G)$. Then by means of combinatorial properties of this graph, we discuss the structure of the group $G$. In particular, we consider the cases where $B(G)$ is a path or a cycle.

math.GR↗

arXiv2013

Groups having complete bipartite divisor graphs for their conjugacy class sizes

Given a finite group G, the bipartite divisor graph for its conjugacy class sizes is the bipartite graph with bipartition consisting of the set of conjugacy class sizes of G-Z (where Z denotes the centre of G) and the set of prime numbers that divide these conjugacy class sizes, and with {p,n} being an edge if gcd(p,n)\neq 1. In this paper we construct infinitely many groups whose bipartite divisor graph for their conjugacy class sizes is the complete bipartite graph K_{2,5}, giving a solution to a question of Taeri.

math.GR↗