Related papers: Perfect codes in quartic Cayley graphs of generali…

Let $G$ be a $2$-generated group. The generating graph $\Gamma(G)$ is the graph whose vertices are the elements of $G$ and where two vertices $g_1$ and $g_2$ are adjacent if $G = \langle g_1, g_2 \rangle.$ This graph encodes the…

The Star graph $S_n$ is the Cayley graph of the symmetric group $Sym_n$ with the generating set $\{(1\mbox{ }i): 2\leq i\leq n \}$. Arumugam and Kala proved that $\{\pi\in Sym_n: \pi(1)=1\}$ is a perfect code in $S_n$ for any $n, n\geq 3$.…

For any positive integer $k$, let $\mathcal{G}_k$ denote the set of finite groups $G$ such that all Cayley graphs ${\rm Cay}(G,S)$ are integral whenever $|S|\le k$. Est${\rm \acute{e}}$lyi and Kov${\rm \acute{a}}$cs \cite{EK14} classified…

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 co-maximal subgroup graph $\Gamma(G)$ of a group $G$ is a graph whose vertices are non-trivial proper subgroups of $G$ and two vertices $H$ and $K$ are adjacent if $HK=G$. In this paper, we continue the study of $\Gamma(G)$, especially…

Quantum symmetry of graph $C^{*}$-algebras has been studied, under the consideration of different formulations, in the past few years. It is already known that the compact quantum group $(\underbrace{C(S^{1})*C(S^{1})*\cdots…

In this communication, the co-maximal subgroup graph $\Gamma(G)$ of a finite group $G$ is examined when $G$ is a finite nilpotent group, finite abelian group, dihedral group $D_n$, dicyclic group $Q_{2^n}$, and $p$-group. We derive the…

A graph $G$ is called well-covered if all maximal independent sets of vertices have the same cardinality. A simplicial complex $\Delta$ is called pure if all of its facets have the same cardinality. Let $\mathcal G$ be the class of graphs…

For a group $G$ and a subset $X$ of $G$, the commuting graph of $X$, denoted by $\Gamma(G,X)$ is the graph whose vertex set is $X$ and any two vertices $u$ and $v$ in $X$ are adjacent if and only if they commute in $G$. In this article,…

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…

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 us say that a Cayley graph $\Gamma$ of a group $G$ of order $n$ is a Cerny Cayley graph if every synchronizing automaton containing $\Gamma$ as a subgraph with the same vertex set admits a synchronizing word of length at most $(n-1)^2$.…

Given a finite simple graph $\Gamma$ on $n$ vertices its complementary prism is the graph $\Gamma\bar{\Gamma}$ that is obtained from $\Gamma$ and its complement $\bar{\Gamma}$ by adding a perfect matching, where each its edge connects two…

Fractional matching extendability is a concept that brings together two widely studied topics in graph theory, namely that of fractional matchings and that of matching extendability. A {\em fractional matching} of a graph $\Gamma$ with edge…

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…

We present an algebraic characterization of perfect graphs, i.e., graphs for which the clique number and the chromatic number coincide for every induced subgraph. We show that a graph is perfect if and only if certain nonnegative…

Let $\Gamma$ be a $G$-symmetric graph with vertex set $V$. We suppose that $V$ admits a $G$-partition $\mathcal{B} = \{ B_0, ... , B_b \}$, with parts of size $v$, and that the quotient graph induced on $\mathcal B$ is a complete graph of…

Let $G$ be a finite group and let $S$ be an inverse-closed subset of $G$ not containing the identity. The Cayley graph $\mathrm{Cay}(G,S)$ has vertex set $G$, where two vertices $x$ and $y$ are adjacent if and only if $x^{-1}y \in S$.…

An equitable partition of a graph $\Ga$ is a partition $\{V_1, \ldots, V_m\}$ of its vertex set such that for each pair $i, j$ all vertices in $V_i$ have the same number of neighbours in $V_j$. When $m=2$, $V_1$ is called an $(a,…

Let $G$ be a finite group. For some fixed prime $p$, let $\Gamma_p(G)$ be the common divisor graph built on the set of sizes of $p$-regular conjugacy classes of $G$: this is the simple undirected graph whose vertices are the class sizes of…