Searcharxiv⌕ Search

arXiv subjects

Alireza Abdollahi

Publications and source records attributed to Alireza Abdollahi.

At least 19 recordsLinked to original sources
arXiv2026

Finite groups with quadratic splitting fields for all Cayley graphs

For a graph $Γ$, the splitting field of $Γ$ is defined as the splitting field of the characteristic polynomial of $Γ$ over rationals. The algebraic degree of $Γ$ is defined by the extension degree of its splitting field over rationals. Let $k$ be a positive integer. We call a finite group $G$ \textit{Cayley $k$-integral} if, for every inverse-closed subset $S$ of $G$, the algebraic degree of the Cayley graph $\Cay(G,S)$ does not exceed $k$. We give a complete classification of all finite Cayley $2$-integral groups. It is shown that a finite abelian group is Cayley $2$-integral if and only if it is isomorphic to one of the following forms: $G \cong \mathbb{Z}_2^r \times \mathbb{Z}_5^s$, $\mathbb{Z}_2^r \times \mathbb{Z}_4^s \times \mathbb{Z}_8^t$, or $\mathbb{Z}_2^r \times \mathbb{Z}_3^s \times \mathbb{Z}_{12}^t$, where $r, s, t \geq 0$. Furthermore, we prove that the set of finite non-abelian Cayley $2$-integral groups consists of the infinite family $Q_8 \times \mathbb{Z}_2^n$, with $n \geq 0$, and $22$ specific groups.

math.CO↗
arXiv2025

Perfect codes and regular sets in vertex-transitive graphs

A subset \( C \) of the vertex set \( V \) of a graph \( Γ= (V,E) \) is termed an $(r,s)$-regular set if each vertex in \( C \) is adjacent to exactly \( r \) other vertices in \( C \), while each vertex not in \( C \) is adjacent to precisely \( s \) vertices in \( C \). A specific case, known as a $(0,1)$-regular set, is referred to as a perfect code. In this paper, we will delve into $(r,s)$-regular sets in the context of vertex-transitive graphs. It is noteworthy that any vertex-transitive graph can be represented as a coset graph \( \Cos(G,H,U) \). When examining a group \( G \) and a subgroup \( H \) of \( G \), a subgroup \( A \) that encompasses \( H \) is identified as an $(r,s)$-regular set related to the pair \( (G,H) \) if there exists a coset graph \( \Cos(G,H,U) \) such that the set of left cosets of \( H \) in \( A \) forms an $(r,s)$-regular set within this graph. In this paper, we present both a necessary and sufficient condition for determining when a normal subgroup \( A \) that includes \( H \) as a normal subgroup qualifies as an $(r,s)$-regular set for the pair \( (G,H) \). Furthermore, if \( A \) is a normal subgroup of \( G \) containing \( H \), we establish a relationship between \( A \) being a perfect code of \( (G,H) \) and the quotient \( N_A(H)/H \) being a perfect code of \(( N_G(H)/H, {1_{N_{G}(H)/H}}) \).

math.CO↗
arXiv2025

On $2$-integral Cayley graphs

In this paper, we introduce the concept of $k$-integral graphs. A graph $Γ$ is called $k$-integral if the extension degree of the splitting field of the characteristic polynomial of $Γ$ over rational field $\mathbb Q$ is equal to $k$. We prove that the set of all finite connected graphs with given algebraic degree and maximum degree is finite. $1$-integral graphs are just integral ones, graphs all of whose eigenvalues are integer. We study $2$-integral Cayley graphs over finite groups $G$ with respect to Cayley sets which are a union of conjugacy classes of $G$. Among other general results, we completely characterize all finite abelian groups having a connected $2$-integral Cayley graph with valency $2,3,4$ and $5$. Furthermore, we classify finite groups $G$ for which all Cayley graphs over $G$ with bounded valency are $2$-integral.

math.CO↗
arXiv2024

Improved bounds on the size of permutation codes under Kendall $τ$-metric

In order to overcome the challenges caused by flash memories and also to protect against errors related to reading information stored in DNA molecules in the shotgun sequencing method, the rank modulation is proposed. In the rank modulation framework, codewords are permutations. In this paper, we study the largest size $P(n, d)$ of permutation codes of length $n$, i.e., subsets of the set $S_n$ of all permutations on $\{1,\ldots, n\}$ with the minimum distance at least $d\in\{1,\ldots ,\binom{n}{2}\}$ under the Kendall $τ$-metric. By presenting an algorithm and some theorems, we managed to improve the known lower and upper bounds for $P(n,d)$. In particular, we show that $P(n,d)=4$ for all $n\geq 6$ and $\frac{3}{5}\binom{n}{2}< d \leq \frac{2}{3} \binom{n}{2}$. Additionally, we prove that for any prime number $n$ and integer $r\leq \frac{n}{6}$, $ P(n,3)\leq (n-1)!-\dfrac{n-6r}{\sqrt{n^2-8rn+20r^2}}\sqrt{\dfrac{(n-1)!}{n(n-r)!}}. $ This result greatly improves the upper bound of $P(n,3)$ for all primes $n\geq 37$.

cs.IT↗
arXiv2023

Integral Cayley graphs of symmetric groups on transpositions

We study subsets $T$ consisting of some transpositions $(i,j)$ of the symmetric group $S_n$ on $\{1,\dots,n\}$ such that the Cayley graph $Γ_T:=Cay(S_n,T)$ is an integral graph, i.e., all eigenvalues of an adjacency matrix of $Γ_T$ are integers. Graph properties of $Γ_T$ are determined in terms of ones of the graph $G_T$ whose vertex set is $\{1,\dots,n\}$ and $\{i,j\}$ is an edge if and only if $(i,j)\in T$. Here we prove that if $G_T$ is a tree then $Γ_T$ is integral if and only if $T$ is isomorphic to the star graph $K_{1,n-1}$, answering Problem 5 of [Electron. J. Comnin., 29(2) (2022) \# P2.9]. Problem 6 of the latter article asks to find necessary and sufficient conditions on $T$ for integralness of $Cay(S_n,T)$ without any further assumption on $T$. We show that if $G_T$ is a graph which we call it a ``generalized complete multipartite graph" then $Cay(S_n,T)$ is integral. We conjecture that $Cay(S_n,T)$ is integral only if $G_T$ is a generalized complete multipartitie graph. To support the latter conjecture we show its validity whenever $G_T$ is some classes of graphs including cycles and cubic graphs.

math.CO↗
arXiv2023

New Upper Bounds on the Size of Permutation Codes under Kendall $τ$-Metric

We first give two methods 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 $S_n$ all permutations on $\{1,\dots,n\}$ 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 $S_n$. The second method can be applied to refute the existence of perfect codes in $S_n$.\\ Here we reduce the known upper bound $(n-1)!-1$ for $P(n,3)$ to $(n-1)!-\lceil\frac{n}{3}\rceil+2\leq (n-1)!-2$, whenever $n\geq 11$ is any prime number. 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.

math.CO↗
arXiv2022

Nilpotent probability of compact groups

Let $k$ be any positive integer and $G$ a compact (Hausdorff) group. Let $\mf{np}_k(G)$ denote the probability that $k+1$ randomly chosen elements $x_1,\dots,x_{k+1}$ satisfy $[x_1,x_2,\dots,x_{k+1}]=1$. We study the following problem: If $\mf{np}_k(G)>0$ then, does there exist an open nilpotent subgroup of class at most $k$? The answer is positive for profinite groups and we give a new proof. We also prove that the connected component $G^0$ of $G$ is abelian and there exists a closed normal nilpotent subgroup $N$ of class at most $k$ such that $G^0N$ is open in $G$.

math.GR↗
arXiv2021

On finite totally 2-closed groups

An abstract group $G$ is called totally $2$-closed if $H=H^{(2),Ω}$ for any set $Ω$ with $G\cong H\leq{\rm Sym}(Ω)$, where $H^{(2),Ω}$ is the largest subgroup of ${\rm Sym}(Ω)$ whose orbits on $Ω\timesΩ$ are the same orbits of $H$. In this paper, we classify the finite soluble totally $2$-closed groups. We also prove that the Fitting subgroup of a totally $2$-closed group is a totally $2$-closed group. Finally, we prove that a finite insoluble totally $2$-closed group $G$ of minimal order with non-trivial Fitting subgroup has shape $Z\cdot X$, with $Z=Z(G)$ cyclic, and $X$ is a finite group with a unique minimal normal subgroup, which is nonabelian.

math.GR↗
arXiv2021

Commuting Probability of Compact Groups

For any (Hausdorff) compact group $G$ with the normalized Haar measure ${\mathbf m}_G$, denote by ${\rm cp}(G)$ the probability ${\mathbf m}_{G\times G}(\{(x,y)\in G\times G \;|\; xy=yx\})$ of commuting a randomly chosen pair of elements of $G$. Here we prove that if ${\rm cp}(G)>0$, then there exists a finite group $H$ such that ${\rm cp}(G)= \frac{{\rm cp}(H)}{|G:F|^2}$, where $F$ is the FC-center of $G$ i.e. the set of all elements of $G$ whose conjugacy classes are finite and $H$ is isoclinic to $F$ with ${\rm cp}(F)={\rm cp}(H)$. The latter equality enables one to transfer many existing results concerning commuting probability of finite groups to one of compact groups. For example, here for a compact group $G$ we prove that if ${\rm cp}(G)>\frac{3}{40}$ then either $G$ is solvable or, else $G\cong A_5 \times T$ for some abelian group $T$, in which case ${\rm cp}(G)=\frac{1}{12}$; where $A_5$ denotes the alternating group of degree $5$.

math.GR↗
arXiv2021

Profinite groups with many elements of bounded order

Lévai and Pyber proposed the following as a conjecture: Let $G$ be a profinite group such that the set of solutions of the equation $x^n=1$ has positive Haar measure. Then $G$ has an open subgroup $H$ and an element $t$ such that all elements of the coset $tH$ have order dividing $n$ (see Problem 14.53 of [The Kourovka Notebook, No. 19, 2019]). \\ We define a constant $c_n$ for all finite groups and prove that the latter conjecture is equivalent with a conjecture saying $c_n<1$. Using the latter equivalence we observe that correctness of Lévai and Pyber conjecture implies the existence of the universal upper bound $\frac{1}{1-c_n}$ on the index of generalized Hughes-Thompson subgroup $H_n$ of finite groups whenever it is non-trivial. It is known that the latter is widely open even for all primes $n=p\geq 5$. For odd $n$ we also prove that Lévai and Pyber conjecture is equivalent to show that $c_n$ is less than $1$ whenever $c_n$ is only computed on finite solvable groups. \\ The validity of the conjecture has been proved in [Arch. Math. (Basel) 75 (2000) 1-7] for $n=2$. Here we confirm the conjecture for $n=3$.

math.GR↗
arXiv2021

Compact groups with a set of positive Haar measure satisfying a nilpotent law

The following question is proposed in [4, Question 1.20]: Let $G$ be a compact group, and suppose that $$\mathcal{N}_k(G) = \{(x1,\dots,x_{k+1}) \in G^{k+1} \;\|; [x_1,\dots, x_{k+1}] = 1\}$$ has positive Haar measure in $G^{k+1}$. Does $G$ have an open $k$-step nilpotent subgroup? The case $k = 1$ is already known. We positively answer it for $k = 2$.

math.GR↗
arXiv2020

A conjecture about spectral distances between cycles, paths and certain trees

We confirm the following conjecture which has been proposed in [{\em Linear Algebra and its Applications}, {\bf 436} (2012), No. 5, 1425-1435.]: $$ 0.945\approx\displaystyle\lim_{n\longrightarrow \infty}σ(P_n,Z_n)=\displaystyle\lim_{n\longrightarrow \infty}σ(W_n,Z_n)=\frac{1}{2}\displaystyle\lim_{n\longrightarrow \infty}σ(P_n,W_n);\ \displaystyle\lim_{n\longrightarrow \infty}σ(C_{2n},Z_{2n})=2,$$ where $σ(G_1,G_2)=\sum_{i=1}^n |λ_i(G_1)-λ_i(G_2)|$ is the spectral distance between $n$ vertex non-isomorphic graphs $G_1$ and $G_2$ with adjacency spectra $λ_1(G_i) \geq λ_2(G_i) \geq \cdots \geq λ_n(G_i)$ for $i=1,2$, and $P_n$ and $C_n$ denote the path and cycle on $n$ vertices, respectively; $Z_n$ denotes the coalescence of $P_{n-2}$ and $P_3$ on one of the vertices of degree 1 of $P_{n-2}$ and the vertex of degree $2$ of $P_3$; and $W_n$ denotes the coalescence of $Z_{n-2}$ and $P_3$ on the vertex of degree 1 of $Z_{n-2}$ which is adjacent to a vertex of degree $2$ and the vertex of degree $2$ of $P_3$.

math.CO↗
arXiv2020

Compact groups with many elements of bounded order

Lévai and Pyber proposed the following as a conjecture: Let $G$ be a profinite group such that the set of solutions of the equation $x^n=1$ has positive Haar measure. Then $G$ has an open subgroup $H$ and an element $t$ such that all elements of the coset $tH$ have order dividing $n$ (see Problem 14.53 of [The Kourovka Notebook, No. 19, 2019]). The validity of the conjecture has been proved in [Arch. Math. (Basel) 75 (2000) 1-7] for $n=2$. Here we study the conjecture for compact groups $G$ which are not necessarily profinite and $n=3$; we show that in the latter case the group $G$ contains an open normal $2$-Engel subgroup.

math.GR↗
arXiv2019

$\ell^1$-Cospectrality of graphs

The following problem has been proposed in [Research problems from the Aveiro workshop on graph spectra, {\em Linear Algebra and its Applications}, {\bf 423} (2007) 172-181.]:\\ (Problem AWGS.4) Let $G_n$ and $G'_n$ be two nonisomorphic graphs on $n$ vertices with spectra $$λ_1 \geq λ_2 \geq \cdots \geq λ_n \;\;\;\text{and}\;\;\; λ'_1 \geq λ'_2 \geq \cdots \geq λ'_n,$$ respectively. Define the distance between the spectra of $G_n$ and $G'_n$ as $$λ(G_n,G'_n) =\sum_{i=1}^n (λ_i-λ'_i)^2 \;\;\; \big(\text{or use}\; \sum_{i=1}^n|λ_i-λ'_i|\big).$$ %Let $ε$ be a nonnegative number. Graphs $G_n$ and $G'_n$ are $ε$-cospectral if $λ(G_n,G'_n)\leq ε$. Thus, $G_n$ %and $G'_n$ are $0$-cospectral if and only if $G_n$ and $G'_n$ are cospectral. Define the cospectrality of $G_n$ by $$\text{cs}(G_n) = \min\{λ(G_n,G'_n) \;:\; G'_n \;\;\text{not isomorphic to} \; G_n\}.$$ %Thus $\text{cs}(G_n) = 0$ if and only if $G_n$ has a cospectral mate. %This function measures how far apart the spectrum of a graph with $n$ vertices can be from the %spectrum of any other graph with $n$ vertices.\\ {\bf Problem A.} Investigate $\text{cs}(G_n)$ for special classes of graphs. In this paper we study Problem A for certain graphs with respect to the $\ell^1$-norm, i.e. $σ(G_n,G'_n)=\sum_{i=1}^n|λ_i-λ'_i|$. We find $\text{cs}(K_n)$, $\text{cs}(nK_1)$, $\text{cs}(K_2+(n-2)K_1)$ ($n\geq 2$), $\text{cs}(K_{n,n})$ and $\text{cs}(K_{n,n+1})$, where $K_n, nK_1, K_2+(n-2)K_1, K_{n,m} $ denote the complete graph on $n$ vertices, the null graph on $n$ vertices, the disjoint union of the $K_2$ with $n-2$ isolated vertices ($n\geq 2$), and the complete bipartite graph with parts of sizes $n$ and $m$, respectively.

math.CO↗
arXiv2019

Zero divisors of support size $3$ in group algebras and trinomials divided by irreducible polynomials over $GF(2)$

A famous conjecture about group algebras of torsion-free groups states that there is no zero divisor in such group algebras. A recent approach to settle the conjecture is to show the non-existence of zero divisors with respect to the length of possible ones, where by the length we mean the size of the support of an element of the group algebra. The case length $2$ cannot be happen. The first unsettled case is the existence of zero divisors of length $3$. Here we study possible length $3$ zero divisors in rational group algebras and in the group algebras over the field with $p$ elements for some prime $p$.

math.GR↗
arXiv2019

Cardinality of product sets in torsion-free groups and applications in group algebras

Let $G$ be a unique product group, i.e., for any two finite subsets $A$ and $B$ of $G$ there exists $x\in G$ which can be uniquely expressed as a product of an element of $A$ and an element of $B$. We prove that, if $C$ is a finite subset of $G$ containing the identity element such that $\langle C\rangle$ is not abelian, then for all subsets $B$ of $G$ with $|B|\geq 7$, $|BC|\geq |B| + |C| + 2$. Also, we prove that if $C$ is a finite subset containing the identity element of a torsion-free group $G$ such that $|C| = 3$ and $\langle C\rangle$ is not abelian, then for all subsets $B$ of $G$ with $|B|\geq 7$, $|BC|\geq |B| + 5$. Moreover, if $\langle C\rangle$ is not isomorphic to the Klein bottle group, i.e., the group with the presentation $\langle x, y \;|\; xyx = y\rangle$, then for all subsets $B$ of G with $|B|\geq 5$, $|BC|\geq |B| + 5$. The support of an element $α=\sum_{x\in G} α_x x$ a group algebra $F[G]$ ($F$ is any field), denoted by $supp(α)$, is the set $\{x\in G \;|\; α_x \neq 0\}$. By the latter result, we prove that if $αβ= 0$ for some non-zero $α,β\in F[G]$ such that $|supp(α)| = 3$, then $|supp(β)|\geq 12$. Also, we prove that if $αβ= 1$ for some $αβ\in F[G]$ such that $|supp(α)| = 3$, then |supp(α)|\geq 10$. These results improve a part of results in Schweitzer [J. Group Theory, 16 (2013), no. 5, 667-693] and Dykema et al. [Exp. Math., 24 (2015), 326-338] to arbitrary fields, respectively.

math.GR↗