Related papers: Regular Cayley maps on dihedral groups with the sm…

In this paper, we provide the first known infinite family of simple graphs, each of which is the skeleton of a chiral map, a skeleton of a reflexible map on an orientable surfaces, as well as a skeleton of a reflexible map on a…

Let $m$ be a positive integer. A group $G$ is said to be an $m$-DCI-group or an $m$-CI-group if $G$ has the $k$-DCI property or $k$-CI property for all positive integers $k$ at most $m$, respectively. Let $G$ be a dihedral group of order…

An orientation $D$ of a graph $G=(V,E)$ is a digraph obtained from $G$ by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each $v \in V(G)$, the indegree of $v$ in $D$, denoted by $d^-_D(v)$, is…

In this note, we introduce a family of "power sum" kernels and the corresponding Gaussian processes on symmetric groups $\mathrm{S}_n$. Such processes are bi-invariant: the action of $\mathrm{S}_n$ on itself from both sides does not change…

For a finite group $G$, the size of a minimum generating set of $G$ is denoted by $d(G)$. Given a finite group $G$ and an integer $k$, deciding if $d(G)\leq k$ is known as the minimum generating set (MIN-GEN) problem. A group $G$ of order…

Let $D=(V,A)$ be a digraph and consider an arc subset $A'\subseteq A$ and an exhaustive mapping $\phi: A\to A'$ such that $(i)$ the set of heads of $A'$ is $H(A')=V$; $(ii)$ the map fixes the elements of $A'$, that is, $\phi|A'=Id$, and for…

Let $D$ be a digraph. We call a subset $N$ of $V(D)$ $k$-independent if for every pair of vertices $u,v \in N$, $d(u,v) \geq k$; and we call it $\ell$-absorbent if for every vertex $u \in V(D) \setminus N$, there exists $v \in N$ such that…

A theoretical framework is established for explicitly calculating rigid kernels of self-similar regular branch groups. This is applied to a new infinite family of branch groups in order to provide the first examples of self-similar, branch…

A finite group $R$ is a DCI-group if, whenever $S$ and $T$ are subsets of $R$ with the Cayley graphs ${\rm Cay}(R,S)$ and ${\rm Cay}(R,T)$ isomorphic, there exists an automorphism $\varphi$ of $R$ with $S^\varphi=T$. Elementary abelian…

In this paper we consider the critical group of finite connected graphs which admit harmonic actions by the dihedral group $D_n$, extending earlier work by the author and Criel Merino. In particular, we show that the critical group of such…

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 map is a cellular decomposition of a closed surface. In the framework of classifying all regular maps by their supporting surface, it is an open problem to find all closed surfaces that support no regular maps. Classification of regular…

For any finite group $G$, a natural question to ask is the order of the smallest possible automorphism group for a Cayley graph on $G$. A particular Cayley graph whose automorphism group has this order is referred to as an MRR (Most Rigid…

Let $k$ be an integer with $k\geq 2$. A $k$-king in a digraph $D$ is a vertex which can reach every other vertex by a directed path of length at most $k$ and a non-king is a vertex which is not a 3-king. A subset $K$ is $k$-independent if…

In this paper we study the diameter of Inhomogeneous random graphs $G(n,\kappa,p)$ that are induced by irreducible kernels $\kappa$. The kernels we consider act on separable metric spaces and are almost everywhere continuous. We generalize…

In this paper, we establish an integral expression for the Dunkl kernel in the context of Dihedral group of an arbitrary order by using the results in \cite{M-Y-Vk} where a construction of the Dunkl intertwining operator for a large set of…

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…

We consider the Minimum Coverage Kernel problem: given a set $B$ of $d$-dimensional boxes, find a subset of $B$ of minimum size covering the same region as $B$. This problem is $\mathsf{NP}$-hard, but as for many $\mathsf{NP}$-hard problems…

The $Kepler$ $orbits$ form a 3-parameter family of $unparametrized$ plane curves, consisting of all conics sharing a focus at a fixed point. We study the geometry and symmetry properties of this family, as well as natural 2-parameter…

Let $G$ be a generalized dicyclic group with identity $1$. An inverse closed subset $S$ of $G\setminus\{1\}$ is called minimal if $\langle S\rangle=G$ and there exists some $s\in S$ such that $\langle S\setminus\{s,s^{-1}\} \rangle\neq G$.…