Related papers: Rotational circulant graphs

A finite transitive permutation group is said to be 3/2-transitive if all the nontrivial orbits of a point stabilizer have the same size greater than 1. Examples include the 2-transitive groups, Frobenius groups and several other less…

We define super-Cayley graphs over a finite abelian group $G$. Using the theory of supercharacters on $G$, we explain how their spectra can be realized as a super-Fourier transform of a superclass characteristic function. Consequently, we…

A finite non-abelian group $H$ is hamiltonian if all of its subgroups are normal. We compute the minimal orders of graphs having a hamiltonian group as their automorphism group. The fixing number of a graph $\Gamma$ is the minimum…

It is shown, for a given graph group $G$, that the fixed point subgroup Fix$\,\varphi$ is finitely generated for every endomorphism $\varphi$ of $G$ if and only if $G$ is a free product of free abelian groups. The same conditions hold for…

The commuting graph of a group $G$ is the graph whose vertices are the elements of $G$, two distinct vertices joined if they commute. Our purpose in this paper is twofold: we discuss the computational problem of deciding whether a given…

Let $G$ be a finite, non-abelian group of the form $G = A N$, where $A \leq G$ is abelian, and $N \trianglelefteq G$ is cyclic. We prove that the commuting graph $\Gamma(G)$ of $G$ is either a connected graph of diameter at most four, or…

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…

It has long been known that a vertex-transitive graph $\Gamma$ is isomorphic to a double coset graph $\text{Cos}(G,H,S)$ of a transitive group $G\le\text{Aut}(\Gamma)$, a vertex stabilizer $H\le G$, and some subset $S\subseteq G$. We show…

Consider the random Cayley graph of a finite group $G$ with respect to $k$ generators chosen uniformly at random, with $1 \ll k \lesssim \log |G|$. The results of this article supplement those in the three main papers on random Cayley…

We introduce shortcut graphs and groups. Shortcut graphs are graphs in which cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly…

Flip graphs are graphs on combinatorial objects in which the adjacency relation reflects a local change in the underlying objects. In this thesis we introduce Yoke graphs, a family of flip graphs that generalizes previously studied families…

In the paper the foundation of the $k$-orbit theory is developed. The theory opens a new simple way to the investigation of groups and multidimensional symmetries. The relations between combinatorial symmetry properties of a $k$-orbit and…

A rank 3 graph is an orbital graph of a rank 3 permutation group of even order. Despite the classification of rank 3 graphs being complete, see, e.g., Chapter 11 of the recent monograph 'Strongly regular graphs' by Brouwer and Van…

A Cayley graph $\Cay(G,S)$ is said to be inner-automorphic if $S$ is a union of conjugacy classes of a group $G$, and arc-transitive if its full automorphism group acts transitively on the set of arcs. In this paper, we characterize four…

In this paper, generalized Cayley graphs are studied. It is proved that every generalized Cayley graph of order 2p is a Cayley graph, where p is a prime. Special attention is given to generalized Cayley graphs on Abelian groups. It is…

We give a unified approach to analysing, for each positive integer $s$, a class of finite connected graphs that contains all the distance transitive graphs as well as the locally $s$-arc transitive graphs of diameter at least $s$. A graph…

Peano partial cubes are the bipartite graphs whose geodesic interval spaces are (closed) join spaces. They are the partial cubes all of whose finite convex subgraphs have a pre-hull number which is at most 1. Special Peano partial cubes are…

Given a finite group $G$, its prime graph $\Gamma(G)$ (also known as its Gruenberg-Kegel graph) is the graph whose vertices are the prime divisors of $|G|$ and where edges $\{p, q\}$ exist whenever $G$ contains an element of order $pq$. We…

In 1982, Durnberger proved that every connected Cayley graph of a finite group with a commutator subgroup of prime order contains a hamiltonian cycle. In this paper, we extend this result to the infinite case. Additionally, we generalize…

A graph $\G$ is {\em symmetric} or {\em arc-transitive} if its automorphism group $\Aut(\G)$ is transitive on the arc set of the graph, and $\G$ is {\em basic} if $\Aut(\G)$ has no non-trivial normal subgroup $N$ such that the quotient…