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

Finite 33-connected homogeneous graphs

A finite graph \G is said to be {\em $(G,3)$-$($connected$)$ homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most $3$ extends to an automorphism $g\in G$ of the graph, where $G$ is a group of automorphisms of the graph. In 1985, Cameron and Macpherson determined all finite $(G, 3)$-homogeneous graphs. In this paper, we develop a method for characterising $(G,3)$-connected homogeneous graphs. It is shown that for a finite $(G,3)$-connected homogeneous graph \G=(V, E), either G_v^{\G(v)} is $2$--transitive or G_v^{\G(v)} is of rank $3$ and \G has girth $3$, and that the class of finite $(G,3)$-connected homogeneous graphs is closed under taking normal quotients. This leads us to study graphs where $G$ is quasiprimitive on $V$. We determine the possible quasiprimitive types for $G$ in this case and give new constructions of examples for some possible types

Finite ss-geodesic-transitive digraphs

This paper initiates the investigation of the family of $(G,s)$-geodesic-transitive digraphs with $s\geq 2$. We first give a global analysis by providing a reduction result. Let $\Gamma$ be such a digraph and let $N$ be a normal subgroup of $G$ maximal with respect to having at least $3$ orbits. Then the quotient digraph $\Gamma_N$ is $(G/N,s')$-geodesic-transitive where s'=\min\{s,\diam(\Gamma_N)\}, $G/N$ is either quasiprimitive or bi-quasiprimitive on $V(\Gamma_N)$, and $\Gamma_N$ is either directed or an undirected complete graph. Moreover, it is further shown that if $\Gamma$ is not $(G,2)$-arc-transitive, then $G/N$ is quasiprimitive on $V(\Gamma_N)$. On the other hand, we also consider the case that the normal subgroup $N$ of $G$ has one orbit on the vertex set. We show that if $N$ is regular on $V(\Gamma)$, then $\Gamma$ is a circuit, and particularly each $(G,s)$-geodesic-transitive normal Cayley digraph with $s\geq 2$, is a circuit. Finally, we investigate $(G,2)$-geodesic-transitive digraphs with either valency at most 5 or diameter at most 2. Let $\Gamma$ be a $(G,2)$-geodesic-transitive digraph. It is proved that: if $\Gamma$ has valency at most $5$, then $\Gamma$ is $(G,2)$-arc-transitive; if $\Gamma$ has diameter $2$, then $\Gamma$ is a balanced incomplete block design with the Hadamard parameters

Characterising vertex-star transitive and edge-star transitive graphs

Recent work of Lazarovich provides necessary and sufficient conditions on a graph L for there to exist a unique simply-connected (k, L)-complex. The two conditions are symmetry properties of the graph, namely vertex-star transitivity and edge-star transitivity. In this paper we investigate vertex- and edge-star transitive graphs by studying the structure of the vertex and edge stabilisers of such graphs. We also provide new examples of graphs that are both vertex-star transitive and edge-star transitive