Related papers: Maps, simple groups, and arc-transitive graphs

We investigate connected cubic vertex-transitive graphs whose edge sets admit a partition into a $2$-factor $\mathcal{C}$ and a $1$-factor that is invariant under a vertex-transitive subgroup of the automorphism group of the graph and where…

We describe computational results about undirected graphs having $14$ vertices and automorphism group isomorphic to $\mathbb{Z}/8\mathbb{Z}$, graphs $\Gamma$ which have less than $2|Aut(\Gamma)|$ vertices. We give one example of such…

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…

Frucht showed that, for any finite group $G$, there exists a cubic graph such that its automorphism group is isomorphic to $G$. For groups generated by two elements we simplify his construction to a graph with fewer nodes. In the general…

A finite graph $\Gamma$ is called $G$-symmetric if $G$ is a group of automorphisms of $\Gamma$ which is transitive on the set of ordered pairs of adjacent vertices of $\Gamma$. We study a family of symmetric graphs, called the unitary…

We classify those 2-groups G which factorise as a product of two disjoint cyclic subgroups A and B, transposed by an automorphism of order 2. The case where G is metacyclic having been dealt with elsewhere, we show that for each e>2 there…

Let $G$ be a group. \textit{The permutability graph of cyclic subgroups of $G$}, denoted by $\Gamma_c(G)$, is a graph with all the proper cyclic subgroups of $G$ as its vertices and two distinct vertices in $\Gamma_c(G)$ are adjacent if and…

A detailed proof is given of a theorem describing the centraliser of a transitive permutation group, with applications to automorphism groups of objects in various categories of maps, hypermaps, dessins, polytopes and covering spaces, where…

Let $\Gamma$ be a graph and let $G$ be a group of automorphisms of $\Gamma$. The graph $\Gamma$ is called $G$-normal if $G$ is normal in the automorphism group of $\Gamma$. Let $T$ be a finite non-abelian simple group and let $G = T^l$ with…

We define a family of groups that include the mapping class group of a genus g surface with one boundary component and the integral symplectic group Sp(2g,Z). We then prove that these groups are finitely generated. These groups, which we…

Let ${\cal M}$ be a map with the underlying graph $\Gamma$. The automorphism group $Aut({\cal M})$ induces a natural action on the set of all vertex-edge-face incident triples, called {\em flags} of ${\cal M}$. The map ${\cal M}$ is said to…

The theory of voltage graphs has become a standard tool in the study graphs admitting a semiregular group of automorphisms. We introduce the notion of a cyclic generalised voltage graph to extend the scope of this theory to graphs admitting…

We prove that for every closed, connected, orientable, irreducible 3-manifold, there exists an alternating group A_n which is not the topological symmetry group of any graph embedded in the manifold. We also show that for every finite group…

We study locally closed transformation monoids which contain the automorphism group of the random graph. We show that such a transformation monoid is locally generated by the permutations in the monoid, or contains a constant operation, or…

We give necessary and sufficient conditions for lobe-transitivity of locally finite and locally countable graphs whose connectivity equals 1. We show further that, given any biconnected graph $\Lambda$ and a "code" assigned to each orbit of…

A "folklore conjecture, probably due to Tutte" (as described in [P.D. Seymour, Sums of circuits, Graph theory and related topics (Proc. Conf., Univ. Waterloo, 1977), pp. 341-355, Academic Press, 1979]) asserts that every bridgeless cubic…

For an integer $s\geq1$ and a graph $\Gamma$, a path $(u_0, u_1, \ldots, u_{s})$ composed of vertices of $\Gamma$ is called an {\em $s$-geodesic} if it is a shortest path between $u_0$ and $u_s$. We say that $\Gamma$ is {\em $s$-geodesic…

These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are…

Regular hypermaps with underlying simple hypergraphs are analysed. We obtain an algorithm to classify the regular embeddings of simple hypergraphs with given order, and determine the automorphism groups of regular embedding of simple…

The set $X$ of $k$-subsets of an $n$-set has a natural graph structure where two $k$-subsets are connected if and only if the size of their intersection is $k-1$. This is known as the Johnson graph. The symmetric group $S_n$ acts on the…