Two local conditions on the vertex stabiliser of arc-transitive graphs and their effect on the Sylow subgroups

In this paper we study $G$-arc-transitive graphs $\Delta$ where the permutation group $G_x^{\Delta(x)}$ induced by the stabiliser $G_x$ of the vertex $x$ on the neighbourhood $\Delta(x)$ satisfies the two conditions given in the introduction. We show that for such a $G$-arc-transitive graph $\Delta$, if $(x,y)$ is an arc of $\Delta$, then the subgroup $G_{x,y}^{[1]}$ of $G$ fixing pointwise $\Delta(x)$ and $\Delta(y)$ is a $p$-group for some prime $p$. Next we prove that every $G$-locally primitive (respectively quasiprimitive, semiprimitive) graph satisfies our two local hypotheses. Thus this provides a new Thompson-Wielandt-like theorem for a very large class of arc-transitive graphs. Furthermore, we give various families of $G$-arc-transitive graphs where our two local conditions do not apply and where $G_{x,y}^{[1]}$ has arbitrarily large composition factors

Locally ss-distance transitive graphs

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 is in the class if it is connected and if, for each vertex $v$, the subgroup of automorphisms fixing $v$ acts transitively on the set of vertices at distance $i$ from $v$, for each $i$ from 1 to $s$. We prove that this class is closed under forming normal quotients. Several graphs in the class are designated as degenerate, and a nondegenerate graph in the class is called basic if all its nontrivial normal quotients are degenerate. We prove that, for $s\geq 2$, a nondegenerate, nonbasic graph in the class is either a complete multipartite graph, or a normal cover of a basic graph. We prove further that, apart from the complete bipartite graphs, each basic graph admits a faithful quasiprimitive action on each of its (1 or 2) vertex orbits, or a biquasiprimitive action. These results invite detailed additional analysis of the basic graphs using the theory of quasiprimitive permutation groups.Comment: Revised after referee report

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

Subdegree growth rates of infinite primitive permutation groups

A transitive group $G$ of permutations of a set $\Omega$ is primitive if the only $G$-invariant equivalence relations on $\Omega$ are the trivial and universal relations. If $\alpha \in \Omega$, then the orbits of the stabiliser $G_\alpha$ on $\Omega$ are called the $\alpha$-suborbits of $G$; when $G$ acts transitively the cardinalities of these $\alpha$-suborbits are the subdegrees of $G$. If $G$ acts primitively on an infinite set $\Omega$, and all the suborbits of $G$ are finite, Adeleke and Neumann asked if, after enumerating the subdegrees of $G$ as a non-decreasing sequence $1 = m_0 \leq m_1 \leq ...$, the subdegree growth rates of infinite primitive groups that act distance-transitively on locally finite distance-transitive graphs are extremal, and conjecture there might exist a number $c$ which perhaps depends upon $G$, perhaps only on $m$, such that $m_r \leq c(m-2)^{r-1}$. In this paper it is shown that such an enumeration is not desirable, as there exist infinite primitive permutation groups possessing no infinite subdegree, in which two distinct subdegrees are each equal to the cardinality of infinitely many suborbits. The examples used to show this provide several novel methods for constructing infinite primitive graphs. A revised enumeration method is then proposed, and it is shown that, under this, Adeleke and Neumann's question may be answered, at least for groups exhibiting suitable rates of growth.Comment: 41 page