Related papers: Regular balanced Cayley maps on ${\rm PSL}(2,p)$

A locally checkable labeling problem (LCL) on a group $\Gamma$ asks one to find a labeling of the Cayley graph of $\Gamma$ satisfying a fixed, finite set of "local" constraints. Typical examples include proper coloring and perfect matching…

Motivated by the Benjamini-Schramm non-unicity of percolation conjecture we study the following question. For a given finitely generated non-amenable group $\Gamma$, does there exist a generating set $S$ such that the Cayley graph…

Let $G$ be a group and $S$ an inverse closed subset of $G\setminus \{1\}$. By a Cayley graph $Cay(G,S)$ we mean the graph whose vertex set is the set of elements of $G$ and two vertices $x$ and $y$ are adjacent if $x^{-1}y\in S$. A group…

A subperiodic group is a group of motions of $d$-dimensional Euclidean space $\R^d$ which contains a translation lattice $\Z^r$ of rank $r < d$ as a subgroup of finite index. A classification into abstract group isomorphism classes is…

Colouring problems arising from group-based constructions provide a natural link between combinatorics and algebra, particularly in the study of Cayley graphs and Latin squares. We introduce the notion of colouring bijections of finite…

A subset $C$ of the vertex set of a graph $\Gamma$ is said to be $(a,b)$-regular if $C$ induces an $a$-regular subgraph and every vertex outside $C$ is adjacent to exactly $b$ vertices in $C$. In particular, if $C$ is an $(a,b)$-regular set…

A finite simple graph $\Gamma$ determines a quotient $P_\Gamma$ of the pure braid group, called a graphic arrangement group. We analyze homomorphisms of these groups defined by deletion of sets of vertices, using methods developed in prior…

By a result of Babai, with finitely many exceptions, every group $G$ admits a semi-regular poset representation with three orbits, that is, a poset $P$ with automorphism group $\textrm{Aut}(P) \simeq G$ such that the action of…

We prove a quantitative refinement of the statement that groups of polynomial growth are finitely presented. Let $G$ be a group with finite generating set $S$ and let $\operatorname{Gr}(r)$ be the volume of the ball of radius $r$ in the…

The power graph $\Gamma_G$ of a finite group $G$ is the graph with the vertex set $G$, where two distinct elements are adjacent if one is a power of the other. An $L(2, 1)$-labeling of a graph $\Gamma$ is an assignment of labels from…

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…

As a vital link between group theory and graph theory, Cayley graphs provide a geometric framework for encoding algebraic structures. This study explores the properties of Cayley graphs derived from cyclic groups whose order is the square…

It was proved in [Y.-Q. Feng, C. H. Li and J.-X. Zhou, Symmetric cubic graphs with solvable automorphism groups, {\em European J. Combin.} {\bf 45} (2015), 1-11] that a cubic symmetric graph with a solvable automorphism group is either a…

A graph is called an integral graph when all eigenvalues of its adjacency matrix are integers. We study which Cayley graphs over a nonabelian group $$ T_{8n}=\left\langle a,b\mid a^{2n}=b^8=e,a^n=b^4,b^{-1}ab=a^{-1} \right \rangle $$ are…

We establish for the matrix group $G=\mathrm{SL}_{n}\left(\mathbb{F}_{p}\right)$ that there exist absolute constants $c\in\left(0,1\right)$ and $C>0$ such that any symmetric generating set $A$, with $\left|A\right|\geq\left|G\right|^{1-c}$…

Let $\Gamma(G,S)$ denote the Cayley graph of a group $G$ with respect to a set $S \subset G$. In this paper, we analyze the spectral properties of the Cayley graphs $\mathcal{T}_{m,n,k} = \Gamma(\mathbb{Z}_m \ltimes_k \mathbb{Z}_n, \{(\pm…

A connected graph $\Gamma$ of diameter ${\rm diam}(\Gamma) \ge \ell$ is $\ell$-distance-balanced if $|W_{xy}(\Gamma)|=|W_{yx}(\Gamma)|$ for every $x,y\in V(\Gamma)$ with $d_{\Gamma}(x,y)=\ell$, where $W_{xy}(\Gamma)$ is the set of vertices…

The relative Cayley graph of a group $G$ with respect to its proper subgroup $H$, is a graph whose vertices are elements of $G$ and two vertices $h\in H$ and $g\in G$ are adjacent if $g=hc$ for some $c\in C$, where $C$ is an inversed-closed…

Let $\Gamma$ be a connected $7$-valent symmetric Cayley graph on a finite non-abelian simple group $G$. If $\Gamma$ is not normal, Li {\em et al.} [On 7-valent symmetric Cayley graphs of finite simple groups, J. Algebraic Combin. 56 (2022)…

Mirror graphs were introduced by Bre\v{s}ar et al. in 2004 as an intriguing class of graphs: vertex-transitive, isometrically embeddable into hypercubes, having a strong connection with regular maps and polytope structure. In this article…