Related papers: Critical groups of arithmetical structures on star…

The main goal of this paper is to apply the arithmetic method developed in our previous paper \cite{13} to determine the number of some types of subgroups of finite abelian groups.

A graph is split if there is a partition of its vertex set into a clique and an independent set. The present paper is devoted to the splitness of some graphs related to finite simple groups, namely, prime graphs and solvable graphs, and…

We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with particular emphasis on the set D' comprised of differences between successive…

Let $\gamma(G)$ denote the domination number of a graph $G$. A vertex $v\in V(G)$ is called a \emph{critical vertex} of $G$ if $\gamma(G-v)=\gamma(G)-1$. A graph is called \emph{vertex-critical} if every vertex of it is critical. In this…

We consider the problem of realizing a group as the fundamental group of a graph of groups where the vertex groups are restricted to certain classes (for example, coming from a certain finite list of groups, or having bounded geometric…

Finite groups with very few character values are characterized. The following is the main result of this article: a finite non-abelian group has precisely four character values if and only if it is the generalized dihedral group of a…

We say that finite groups are isospectral if they have the same sets of orders of elements. It is known that every nonsolvable finite group $G$ isospectral to a finite simple group has a unique nonabelian composition factor, that is, the…

Recently, there have been found new relations between the zero forcing number and the minimum rank of a graph with the algebraic co-rank. We continue on this direction by giving a characterization of the graphs with real algebraic co-rank…

Given a finite graph G there is a corresponding group given by the presentation with generators the vertices of G and a relation [x,y]=1 for generators x and y precisely when (x,y) is an edge of G. Such groups are known as partially…

Let $G$ be a group and $S$ an inverse closed subset of $G\setminus \{1\}$. By a Cayley graph $Cay(G,S)$ we mean the graph whose vertex set is the set of elements of $G$ and two vertices $x$ and $y$ are adjacent if $x^{-1}y\in S$. A group…

In this paper, we study the arithmetical structures on Fan Graphs Fn. Let G be a finite and connected graph. An arithmetical structure on G is a pair (d, r) of positive integer vectors such that r is primitive (the greatest common divisor…

We introduce a class of algebras over a field $\mathbb{F}$ related to directed graphs in which all edges are labeled by nonzero elements of the field $\mathbb{F}$. If all labels are different from $1$, these algebras are axial algebras. We…

The set of all centralizers of elements in a finite group $G$ is denoted by $Cent(G)$ and $G$ is called $n-$centralizer if $|Cent(G)| = n$. In this paper, the structure of centralizers in a non-abelian finite group $G$ with this property…

Let $G$ be a finite abelian group of order $n$. For any subset $B$ of $G$ with $B=-B$, the Cayley graph $G_B$ is a graph on vertex set $G$ in which $ij$ is an edge if and only if $i-j\in B.$ It was shown by Ben Green that when $G$ is a…

Critical ideals generalize the critical group, Smith group and the characteristic polynomials of the adjacency and Laplacian matrices of a graph. We give a complete characterization of the digraphs with at most one trivial critical ideal.…

In this paper, the critical group structure of the Cartesian product graph $C_4\times C_n$ is determined, where $n\ge 3$.

A group $G$ has property (VRC) if every cyclic subgroup is a virtual retract. This property is stable under many standard group-theoretic constructions and is enjoyed by all virtually special groups (in the sense of Haglund and Wise). In…

In this work, we classify the group gradings on finite-dimensional incidence algebras over a field, where the field has characteristic zero, or the characteristic is greater than the dimension of the algebra, or the grading group is…

The prime-coprime graph $\Theta(G)$ of a finite group $G$ is the simple graph with vertex set $G$, where two distinct elements are adjacent whenever the greatest common divisor of their orders is either $1$ or a prime. We characterize all…

Let $G$ be a finite group, $S\subseteq G\setminus\{1\}$ be a set such that if $a\in S$, then $a^{-1}\in S$, where $1$ denotes the identity element of $G$. The undirected Cayley graph $Cay(G,S)$ of $G$ over the set $S$ is the graph whose…