Related papers: Generalized quaternion groups with the m-DCI prope…

The prime simplicial complex $\Pi(G)$ of a finite group $G$ is composed of all sets of primes $S$ where $G$ has an element of order the product of primes in $S$, with the subsets partially ordered by inclusion. This complex was introduced…

Let $G$ be a finite abelian group of order $n$ and let $\mathcal M_G=(x_{a+b})_{a,b\in G}$ be the Cayley table of $G$. Let $\text{imm}_\lambda(\mathcal M_G)$ be the immanant of $\mathcal M_G$ with respect to a partition $\lambda$ and…

Let $D$ be a strong digraph on $n\geq 4$ vertices. In [2, J. Graph Theory 22 (2) (1996) 181-187)], J. Bang-Jensen, G. Gutin and H. Li proved the following theorems: If (*) $d(x)+d(y)\geq 2n-1$ and $min \{d(x), d(y)\}\geq n-1$ for every pair…

Let (k1,k2,k3,k4) be a quartet of cyclic cubic number fields sharing a common conductor c=pqr divisible by exactly three prime(power)s p,q,r. For those components k of the quartet whose 3-class group Cl(3,k) = Z/3Z x Z/3Z is elementary…

Let $G$ be a group. Write $G^{*}=G\setminus \{1\}$. An element $x$ of $G^{*}$ will be called deficient if $ \langle x\rangle < C_G(x)$ and it will be called non-deficient if $\langle x\rangle = C_G(x).$ If $x\in G$ is deficient…

For a group $G$ and subsets $S,T \subset G$ we introduce the mirror di-Cayley graph $MX(G;S,T)$ and mirror di-Cayley sum graph $MX^+(G;S,T)$ with connections sets $S$ and $T$ (MDCGs for short). We refer to them indistinctly by…

Let $G$ be a finite group, $n$ a positive integer. $\pi(n)$ denotes the set of all prime divisors of $n$ and $\pi(G)=\pi(|G|)$. The prime graph $\Gamma(G)$ of $G$, defined by Grenberg and Kegel, is a graph whose vertex set is $\pi(G)$, two…

We associate a graph $\Gamma_G$ to a non locally cyclic group $G$ (called the non-cyclic graph of $G$) as follows: take $G\backslash Cyc(G)$ as vertex set, where $Cyc(G)=\{x\in G | \left<x,y\right> \text{is cyclic for all} y\in G\}$, and…

Let $\Gamma=$Cay$(G,T)$ be a Cayley digraph over a finite Abelian group $G$ with respect the generating set $T\not\ni0$. $\Gamma$ has order ord$(\Gamma)=|G|=n$ and degree deg$(\Gamma)=|T|=d$. Let $k(\Gamma)$ be the diameter of $\Gamma$ and…

Let $ 1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$ be an exact sequence of finitely presented groups where Q is infinite and not virtually cyclic, and is the fundamental group of some closed 3-manifold. If G is Kaehler, we…

A $\mathbb{T}$-gain graph is a triple $\Phi=(G,\mathbb{T},\varphi)$ consisting of a graph $G=(V,E)$, the circle group $\mathbb{T}=\{z\in C: |z|=1\}$ and a gain function $\varphi:\overrightarrow{E}\rightarrow \mathbb{T}$ such that…

A Cayley graph of a group $H$ is a finite simple graph $\Gamma$ such that ${\rm Aut}(\Gamma)$ contains a subgroup isomorphic to $H$ acting regularly on $V(\Gamma)$, while a Haar graph of $H$ is a finite simple bipartite graph $\Sigma$ such…

Let $\Gamma$ be an undirected and simple graph. A set $ S $ of vertices in $\Gamma$ is called a {cyclic vertex cutset} of $\Gamma$ if $\Gamma - S$ is disconnected and has at least two components each containing a cycle. If $\Gamma$ has a…

For each pointed abelian group $(A,c)$, there is an associated {\em Galkin quandle} $G(A,c)$ which is an algebraic structure defined on $\Bbb Z_3\times A$ that can be used to construct knot invariants. It is known that two finite Galkin…

Let $D$ be a division ring with infinite center, $n$ be a positive integer and $w(x_1,x_2,\cdots, x_m)=1$ be a generalized group identity over the general linear group $\GL_n(D)$. The aim of this paper is to prove that every subnormal…

Suppose G is a nilpotent, finite group. We show that if {a,b} is any 2-element generating set of G, then the corresponding Cayley digraph Cay(G;a,b) has a hamiltonian path. This implies there is a hamiltonian path in every connected Cayley…

A finitely presented, torsion free, abelian-by-cyclic group can always be written as an ascending HNN extension Gamma_M of Z^n, determined by an n x n integer matrix M with det(M) \ne 0. The group Gamma_M is polycyclic if and only if…

The power graph $\mathscr{P}(G)$ of a group $G$ is defined as the simple graph with vertex set $G$, and where two distinct vertices $x$ and $y$ are joined by an edge if and only if either $x= y^k$ or $y= x^k$, $k \in \mathbb{N}$. Here we…

Let $\Gamma=(V,E)$ be a graph. If all the eigenvalues of the adjacency matrix of the graph $\Gamma$ are integers, then we say that $\Gamma$ is an integral graph. A graph $\Gamma$ is determined by its spectrum if every graph cospectral to it…

Let $G$ be a finite group and $\sigma$ a partition of the set of all? primes $\Bbb{P}$, that is, $\sigma =\{\sigma_i \mid i\in I \}$, where $\Bbb{P}=\bigcup_{i\in I} \sigma_i$ and $\sigma_i\cap \sigma_j= \emptyset $ for all $i\ne j$. If $n$…