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Majid Arezoomand

Publications and source records attributed to Majid Arezoomand.

13 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↗
arXiv2023

Algebraic degrees of quasi-abelian semi-Cayley digraphs

For a digraph $Γ$, if $F$ is the smallest field that contains all roots of the characteristic polynomial of the adjacency matrix of $Γ$, then $F$ is called the splitting field of $Γ$. The extension degree of $F$ over the field of rational numbers $\mathbb{Q}$ is said to be the algebraic degree of $Γ$. A digraph is a semi-Cayley digraph over a group $G$ if it admits $G$ as a semiregular automorphism group with two orbits of equal size. A semi-Cayley digraph $\mathrm{SC}(G,T_{11},T_{22},T_{12},T_{21})$ is called quasi-abelian if each of $T_{11},T_{22},T_{12}$ and $T_{21}$ is a union of some conjugacy classes of $G$. This paper determines the splitting field and the algebraic degree of a quasi-abelian semi-Cayley digraph over any finite group in terms of irreducible characters of groups. This work generalizes the previous works on algebraic degrees of Cayley graphs over abelian groups and any group having a subgroup of index 2, and semi-Cayley digraphs over abelian groups.

math.CO↗
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↗
arXiv2022

Tetravalent vertex-transitive graphs of order $6p$

A graph is vertex-transitive if its automorphism group acts transitively on vertices of the graph. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper, the tetravalent vertex-transitive non-Cayley graphs of order $6p$ are classified for each prime $p$.

math.GR↗
arXiv2022

Perfect state transfer on semi-Cayley graphs over abelian groups

In this paper, we consider the problem on the existence of perfect state transfer(PST for short) on semi-Cayley graphs over abelian groups (which are not necessarily regular), i.e on the graphs having semiregular and abelian subgroups of automorphisms with two orbits of equal size. We stablish a characterization of semi-Cayley graphs over abelian groups having PST. As a result, we give a characterization of Cayley graphs over groups with an abelian subgroup of index 2 having PST, which improves the earlier results on Cayley graphs over abelian groups, dihedral groups and dicyclic group and determines Cayley graphs over generalized dihedral groups and generalized dicyclic groups having PST.

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

Totally $2$-closed finite groups with trivial Fitting subgroup

A group $G$ is said to be totally $2$-closed if in each of its faithful permutation representations, say on a set $Ω$, $G$ is the largest subgroup of $\mathrm{Sym}(Ω)$ which leaves invariant each of the $G$-orbits for the induced action on $Ω\times Ω$. We prove that there are precisely $47$ finite totally $2$-closed groups with trivial Fitting subgroup. Each of these groups is a direct product of pairwise non-isomorphic sporadic simple groups, with the direct factors coming from the Janko groups $\mathrm{J}_1, \mathrm{J}_3$ and $\mathrm{J}_4$, together with $\mathrm{Ly}, \mathrm{Th}$ and the Monster $\mathbb{M}$. These are the first known examples of insoluble totally $2$-closed groups. As a by-product of our methods, we develop several tools for studying $2$-closures of transitive permutation groups -- a vital tool in the study of representations of finite groups as automorphism groups of digraphs. We also prove a dual to a 1939 theorem of Frucht from Algebraic Graph Theory.

math.GR↗
arXiv2021

Fixity of elusive groups and the polycirculant conjecture

Let $G\leq{\rm Sym}(Ω)$ be transitive. Then $G$ is called \textit{elusive} on $Ω$ if it has no fixed point free element of prime order. The \textit{$2$-closure} of $G$, denoted by $G^{(2),Ω}$, is the largest subgroup of ${\rm Sym}(Ω)$ whose orbits on $Ω\timesΩ$ are the same orbits of $G$. $G$ is called $2$-closed on $Ω$ if $G=G^{(2),Ω}$. The \textit{polycirculant conjecture} states that there is no $2$-closed elusive group. In this paper, we study the \textit{fixity} of elusive groups, where the fixity of $G$ is the maximal number of fixed points of a non-trivial element of $G$. In particular, we prove that there is no $2$-closed elusive solvable group of fixity at most $5$, a partial answer to the polycirculant conjecture.

math.GR↗
arXiv2020

Normality of one-matching semi-Cayley graphs over finite abelian groups with maximum degree three

A graph $Γ$ is said to be a semi-Cayley graph over a group $G$ if it admits $G$ as a semiregular automorphism group with two orbits of equal size. We say that $Γ$ is normal if $G$ is a normal subgroup of ${\rm Aut}(Γ)$. We prove that every connected intransitive one-matching semi-Cayley graph, with maximum degree three, over a finite abelian group is normal and characterize all such non-normal graphs.

math.CO↗
arXiv2020

On $\BCI$-groups and $\CI$-groups

Let $G$ be a finite group and $S$ be a subset of $G.$ A bi-Cayley graph $\BCay(G,S)$ is a simple and an undirected graph with vertex-set $G\times\{1,2\}$ and edge-set $\{\{(g,1),(sg,2)\}\mid g\in G, s\in S\}$. A bi-Cayley graph $\BCay(G,S)$ is called a $\BCI$-graph if for any bi-Cayley graph $\BCay(G,T)$, whenever $\BCay(G,S)\cong\BCay(G,T)$ we have $T=gS^σ$ for some $g\in G$ and $σ\in\Aut(G).$ A group $G$ is called a $\BCI$-group if every bi-Cayley graph of $G$ is a $\BCI$-graph. In this paper, we showed that every $\BCI$-group is a $\CI$-group, which gives a positive answer to a conjecture proposed by Arezoomand and Taeri in \cite{arezoomand1}. Also we proved that there is no any non-Abelian $4$-$\BCI$-simple group. In addition all $\BCI$-groups of order $2p$, $p$ a prime, are characterized.

math.GR↗