Related papers: On regular sets in Cayley graphs

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 $\BCay(G,S)$…

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…

We establish a necessary and sufficient condition for a normal subgroup of a finite group to be a subgroup perfect code.

We study two families of cyclotomic graphs and perfect codes in them. They are Cayley graphs on the additive group of $\mathbb{Z}[\zeta_m]/A$, with connection sets $\{\pm (\zeta_m^i + A): 0 \le i \le m-1\}$ and $\{\pm (\zeta_m^i + A): 0 \le…

A vertex coloring of a given simple graph $G=(V,E)$ with $k$ colors ($k$-coloring) is a map from its vertex set to the set of integers $\{1,2,3,\dots, k\}$. A coloring is called perfect if the multiset of colors appearing on the neighbours…

Let $G$ be a finite group and $\text{cd}(G)$ denote the character degree set for $G$. The prime graph $\Delta(G)$ is a simple graph whose vertex set consists of prime divisors of elements in $\text{cd}(G)$, denoted $\rho(G)$. Two primes…

A subset $S$ of a vertex set of a graph $G$ is a total $(k,r)$-dominating set if every vertex $u \in V(G)$ is within distance $k$ of at least $r$ vertices in $S$. The minimum cardinality among all total $(k,r)$-dominating sets of $G$ is…

A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to a subset $S$ of $G$ is called normal if the right regular representation of $G$ is a normal subgroup in the full automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if…

The cage problem asks for the smallest number $c(k,g)$ of vertices in a $k$-regular graph of girth $g$ and graphs meeting this bound are known as cages. While cages are known to exist for all integers $k \ge 2$ and $g \ge 3$, the exact…

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 $H(G,S)$…

A Cayley graph $\Ga=\Cay(G,S)$ is said to be normal if the right-regular representation of $G$ is normal in $\Aut\Ga$. In this paper, we investigate the normality problem of the connected 13-valent symmetric Cayley graphs $\Ga$ of finite…

A set $C$ of vertices of a simple graph is called a completely regular code if for each $i=0$, $1$, $2$, \ldots and $j = i-1$, $i$, $i+1$, all vertices at distance $i$ from $C$ have the same number $s_{ij}$ of neighbors at distance $j$ from…

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 $a,b\in G$…

A subset $D$ of the vertex set $V$ of a graph $G$ is called an $[1,k]$-dominating set if every vertex from $V-D$ is adjacent to at least one vertex and at most $k$ vertices of $D$. A $[1,k]$-dominating set with the minimum number of…

In Cayley graphs on the additive group of a small vector space over GF$(q)$, $q=2,3$, we look for completely regular (CR) codes whose parameters are new in Hamming graphs over the same field. The existence of a CR code in such Cayley graph…

A graph is called integral if all its eigenvalues are integers. A Cayley graph is called normal if its connection set is a union of conjugacy classes. We show that a non-empty integral normal Cayley graph for a group of odd order has an odd…

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…

For groups $G$ that can be generated by an involution and an element of odd prime order, this paper gives a sufficient condition for a certain Cayley graph of $G$ to be a graphical regular representation (GRR), that is, for the Cayley graph…

A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to $S$ is said to be normal if the right regular representation of $G$ is normal in the automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if there is $\alpha\in Aut(G)$…

The Cage Problem requires for a given pair $k \geq 3, g \geq 3$ of integers the determination of the order of a smallest $k$-regular graph of girth $g$. We address a more general version of this problem and look for the $(k,g)$-spectrum of…