Related papers: Regular set in Cayley sum mgraph

Given a finite group $G$ with identity $e$ and a normal subgroup $H$ of $G$, the subgroup sum graph $\Gamma_{G,H}$ (resp. extended subgroup sum graph $\Gamma_{G,H}^+$) of $G$ with respect to $H$ is the graph with vertex set $G$, in which…

A perfect code in a graph $\Gamma=(V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a subgroup…

A subset $C$ of the vertex set of a graph $\Gamma$ is called a perfect code of $\Gamma$ if every vertex of $\Gamma$ is at distance no more than one to exactly one vertex in $C$. In this paper, we classify all connected quintic Cayley graphs…

A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a…

Let $G$ be a finite abelian group, written additively, and $H$ a subgroup of~$G$. The \emph{subgroup sum graph} $\Gamma_{G,H}$ is the graph with vertex set $G$, in which two distinct vertices $x$ and $y$ are joined if $x+y\in…

Given a graph $\Gamma$, a subset $C$ of $V(\Gamma)$ is called a perfect code in $\Gamma$ if every vertex of $\Gamma$ is at distance no more than one to exactly one vertex in $C$, and a subset $C$ of $V(\Gamma)$ is called a total perfect…

A total perfect code in a graph $\Gamma$ is a subset $C$ of $V(\Gamma)$ such that every vertex of $\Gamma$ is adjacent to exactly one vertex in $C$. We give necessary and sufficient conditions for a conjugation-closed subset of a group to…

For a graph $\Gamma=(V\Gamma,E\Gamma)$, a subset $D$ of $V\Gamma$ is a perfect code in $\Gamma$ if every vertex of $\Gamma$ is dominated by exactly one vertex in $D$. In this paper, we classify all connected quartic Cayley graphs on…

Given a graph $\Gamma$, a perfect code in $\Gamma$ is an independent set $C$ of vertices of $\Gamma$ such that every vertex outside of $C$ is adjacent to a unique vertex in $C$, and a total perfect code in $\Gamma$ is a set $C$ of vertices…

Let \( G \) be a finite non-cyclic group. Define \( \mathrm{Cyc}(G) \) as the set of all elements \( a \in G \) such that for any $b\in G$, the subgroup \( \langle a, b \rangle \) is cyclic. The \emph{non-cyclic graph} $\Gamma(G)$ of \( G…

For a digraph $\Gamma$, a subset $C$ of $V(\Gamma)$ is a perfect code if $C$ is a dominating set such that every vertex of $\Gamma$ is dominated by exactly one vertex in $C$. In this paper, we classify strongly connected 2-valent Cayley…

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…

Let $G$ be a group and $S\subseteq G$ its subset such that $S=S^{-1}$, where $S^{-1}=\{s^{-1}\mid s\in S\}$. Then {\it the Cayley graph ${\rm Cay}(G,S)$} is an undirected graph $\Gamma$ with the vertex set $V(\Gamma)=G$ and the edge set…

Let $G$ be a finite group. For each $m>1$ we define the symmetric canonical subset $S=S(m)$ of the Cartesian power $G^m$ and we consider the family of Cayley graphs $\mathscr{G}_m(G)=Cay(G^m,S)$. We describe properties of these graphs and…

Let $G$ denote a finite abelian group with identity 1 and let $S$ denote an inverse-closed subset of $G \setminus {1}$, which generates $G$ and for which there exists $s \in S$, such that $\la S \setminus \{s,s^{-1}\} \ra \ne G$. In this…

A graph $\Gamma$ 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 $\Gamma$ is normal if $G$ is a normal subgroup of ${\rm Aut}(\Gamma)$. We…

Given a finite group $G$ with a normal subgroup $N$, the simple graph $\Gamma_\textit{G}( \textit{N} )$ is a graph whose vertices are of the form $|x^G|$, where $x\in{N\setminus{Z(G)}}$, and $x^G$ is the $G$-conjugacy class of $N$…

Let $G$ be a finite group, and $S$ be a subset of $G\setminus\{1\}$ such that $S=S^{-1}$. Suppose that $Cay(G,S)$ is the Cayley graph on $G$ with respect to the set $S$ which is the graph whose vertex set is $G$ and two vertices $a,b\in G$…

For a finite group $G$ and subset $S$ of $G,$ the Haar graph $H(G,S)$ is a bipartite regular graph, defined as a regular $G$-cover of a dipole with $|S|$ parallel arcs labelled by elements of $S$. If $G$ is an abelian group, then $H(G,S)$…

Inspired by a famous characterization of perfect graphs due to Lov\'{a}sz, we define a graph $G$ to be sum-perfect if for every induced subgraph $H$ of $G$, $\alpha(H) + \omega(H) \geq |V(H)|$. (Here $\alpha$ and $\omega$ denote the…