Related papers: On non-normal subgroup perfect codes

Let G be a finite group and let Irr(G) be the set of all irreducible complex characters of G. Let cd(G) be the set of all character degrees of G and denote by \rho(G) the set of primes which divide some character degrees of G. The prime…

Let G be an exceptional algebraic group defined over an algebraically closed field k of characteristic p>0 and let H be a subgroup of G. Then following Serre we say H is G-completely reducible or G-cr if, whenever H is contained in a…

Consider the following classes of pairs consisting of a group and a finite collection of subgroups: $\mathcal{C}= \left\{ (G,\mathcal H) \mid \text{$\mathcal{H}$ is hyperbolically embedded in $G$} \right\}$ and $ \mathcal{D}= \left\{…

A graph product kernel means the kernel of the natural surjection from a graph product to the corresponding direct product. We prove that a graph product kernel of countable groups is special, and a graph product of finite or cyclic groups…

Let $F=\{H_1,...,H_k\}$ be a family of graphs. A graph $G$ with $m$ edges is called {\em totally $F$-decomposable} if for {\em every} linear combination of the form $\alpha_1 e(H_1) + ... + \alpha_k e(H_k) = m$ where each $\alpha_i$ is a…

A graph $G$ is called \emph{symmetric with respect to a functional $F_G(P)$} defined on the set of all the probability distributions on its vertex set if the distribution $P^*$ maximizing $F_G(P)$ is uniform on $V(G)$. Using the…

For a finite group $G$ and an irreducible complex character $\chi$ of $G$, the codegree of $\chi$ is defined by $\textrm{cod}(\chi)=|G:\textrm{ker}(\chi)|/\chi(1)$, where $\textrm{ker}(\chi)$ is the kernel of $\chi$. In this paper, we show…

Given a graph $G=(V,E)$, a subset $X$ of $V$ is an interval of $G$ provided that for any $a, b\in X$ and $ x\in V \setminus X$, $\{a,x\}\in E$ if and only if $\{b,x\}\in E$. For example, $\emptyset$, $\{x\}(x\in V)$ and $V$ are intervals of…

We give a uniform and self-contained proof that if $G$ is a connected graph with $\chi(G) = \Delta(G)$ and $G\neq \overline{C_7}$, then $G$ contains either $K_{\Delta(G)}$ or an odd hole where every vertex has degree at least $\Delta(G)-1$…

If all the eigenvalues of the Hermitian-adjacency matrix of a mixed graph are integers, then the mixed graph is called \emph{H-integral}. If all the eigenvalues of the (0,1)-adjacency matrix of a mixed graph are \emph{Gaussian integers},…

The complete transposition graph is defined to be the graph whose vertices are the elements of the symmetric group $S_n$, and two vertices $\alpha$ and $\beta$ are adjacent in this graph iff there is some transposition $(i,j)$ such that…

A dominating set $S$ in a graph $G$ is said to be perfect if every vertex of $G$ not in $S$ is adjacent to just one vertex of $S$. Given a vertex subset $S'$ of a side $P_m$ of an $m\times n$ grid graph $G$, the perfect dominating sets $S$…

The Hawkes graph $\Gamma_H(G)$ of $G$ is the directed graph whose vertex set coincides with $\pi(G)$ and it has the edge $(p, q)$ whenever $q\in\pi(G/O_{p',p}(G))$. The Sylow graph $\Gamma_s(G)$ of $G$ is the directed graph with vertex set…

We prove that any finite abelian group $G$ contains a collection of not too many subsets with a special structure, so that for every subset $A$ of $G$ with a small doubling, there is a member $F$ of the collection that is fully contained in…

An \emph{$H$-packing} in a graph $G$ is a collection of pairwise vertex-disjoint copies of $H$ in $G$. We prove that for every $c > 0$ and every bipartite graph $H$, any $\lfloor cn \rfloor$-regular graph $G$ admits an $H$-packing that…

Let $R$ be a commutative ring with unity and $R^{+}$ be $Z^*(R)$ be the additive group and the set of all non-zero zero-divisors of $R$, respectively. We denote by $\mathbb{CAY}(R)$ the Cayley graph $Cay(R^+,Z^*(R))$. In this paper, we…

In this paper, a new graph structure called the \textit{coprime order graph} of a finite group $G$ denoted by $\Theta(G)$ has been introduced. The \textit{coprime graph} of a finite group introduced by Ma, Wei, and Yang [\textit{The coprime…

We construct an infinite family of triples (G,S1, S2) each consisting of a group G and a pair (S1, S2) of distinct subsets of G with the following properties. i The two Cayley graphs Cay(G, S1) and Cay(G,S2) are non-isomorphic. ii The…

Let $G$ be a graph and $\mathcal{H}$ be a hypergraph both on the same vertex set. We say that a hypergraph $\mathcal{H}$ is a \emph{Berge}-$G$ if there is a bijection $f : E(G) \rightarrow E(\mathcal{H})$ such that for $e \in E(G)$ we have…

Let $0<\ell\in\mathbb{Z}$. The notion of an efficient dominating set or perfect code $S$ of a graph $G$ is generalized to that of an efficient dominating$\,^\ell$-set or perfect$^\ell$code, of the graph $G$, meaning that each vertex $v$ of…