Related papers: On oriented $m$-semiregular representations of fin…

In this paper we investigate the $directed$ $normalizing$ $graph$ associated with a group $G$, defined as the simple directed graph whose vertices are the elements of $G$, with an arrow from $x$ to $y$ whenever the subgroup $\langle x…

A finite non-abelian group $G$ is called commuting integral if the commuting graph of $G$ is integral. In this paper, we show that a finite group is commuting integral if its central factor is isomorphic to ${\mathbb{Z}}_p \times…

If $G$ is a group acting on a set $\Omega$ and $\alpha, \beta \in \Omega$, the digraph whose vertex set is $\Omega$ and whose arc set is the orbit $(\alpha, \beta)^G$ is called an {\em orbital digraph} of $G$. Each orbit of the stabiliser…

A finite group $G$ is a called a DCI-group if any two isomorphic Cayley digraphs of $G$ are also isomorphic via an automorphism of $G$. If $G$ is a non-abelian generalised dihedral DCI-group, then Dobson, Muzychuk, and Spiga proved that $G$…

A unitary representation of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i\dd\pi(x)$ from the derived representation are uniformly bounded from above on some non-empty open subset of…

We characterise connected cubic graphs admitting a vertex- transitive group of automorphisms with an abelian normal subgroup that is not semiregular. We illustrate the utility of this result by using it to prove that the order of a…

For semigroup $S$, a commutative congruence $\sigma_{orient}$ on $S$ and a subsemigroup Orientable($S$) of $S$ were introduced in "Two cancellative commutative congruences and group diagrams", Semigroup Forum (2011) 82: 338-353. Here we…

Let $G$ be a finite group of odd order admitting an involutory automorphism $\phi$. We obtain two results bounding the exponent of $[G,\phi]$. Denote by $G_{-\phi}$ the set $\{[g,\phi]\,\vert\, g\in G\}$ and by $G_{\phi}$ the centralizer of…

A \emph{mixed dihedral group} is a group $H$ with two disjoint subgroups $X$ and $Y$, each elementary abelian of order $2^n$, such that $H$ is generated by $X\cup Y$, and $H/H'\cong X\times Y$. In this paper we give a sufficient condition…

We classify the regular maps $\mathcal M$ which have automorphism groups $G$ acting faithfully and primitively on their vertices. As a permutation group $G$ must be of almost simple or affine type, with dihedral point stabilisers. We show…

Let $G$ be a finite group and $P$ a Sylow $2$-subgroup of $G$. We obtain both asymptotic and explicit bounds for the number of odd-degree irreducible complex representations of $G$ in terms of the size of the abelianization of $P$. To do…

We study, from the point of view of CR geometry, the orbits M of a real form G of a complex semisimple Lie group G in a complex flag manifold G/Q. In particular we characterize those that are of finite type and satisfy some Levi…

If $G$ is a group of permutations of a set $\Omega$, then the suborbits of $G$ are the orbits of point-stabilisers $G_\alpha$ acting on $\Omega$. The cardinalities of these suborbits are the subdegrees of $G$. Every infinite primitive…

A graphical regular representation (GRR) of a group $G$ is a Cayley graph of $G$ whose full automorphism group is equal to the right regular permutation representation of $G$. Towards a proof of the conjecture that only finitely many finite…

In this paper we show that almost all Cayley digraphs have automorphism group as small as possible; that is, they are digraphical regular representations (DRRs). More precisely, we show that as $r$ tends to infinity, for every finite group…

A group $G$ is complete group if it satisfies $Z(G)=e$ and $Aut(G)=Inn(G)$. In this paper, on the one hand, we study the basic properties of generalized Cayley graphs and characterize two classes isomorphic generalized generalized Cayley…

We show that finitely presented groups which admit $k$-planar Cayley graphs contain finite-index subgroups with planar Cayley graphs. More generally, we answer a question of Georgakopoulos and Papasoglu in the special case of coarsely…

A discrete analog of quantum unique ergodicity was proved for Cayley graphs of quasirandom groups by Magee, Thomas and Zhao. They show that for large graphs there exist real orthonormal basis of eigenfunctions of the adjacency matrix such…

This paper introduces a notion of presentation for locally inverse semigroups and develops a graph structure to describe the elements of locally inverse semigroups given by these presentations. These graphs will have a role similar to the…

A finite group $G$ is a DCI-group if, whenever $S$ and $S'$ are subsets of $G$ with the Cayley graphs Cay$(G,S)$ and Cay$(G,S')$ isomorphic, there exists an automorphism $\varphi$ of $G$ with $\varphi(S)=S'$. It is a CI-group if this…