### Speaker

Prof.
Marston Conder
(University of Auckland)

### Description

Locally-finite vertex-transitive graphs may be classified according
to the action of the automorphism group on the arcs (ordered edges) of the graph.
Let $X$ be vertex-transitive graph of valency $d$, with full
automorphism group $A$. Then the {\em arc-type\/} of $X$ is defined in terms
of the lengths of the orbits of the action of the stabiliser $A_v$ of a
given vertex $v$ on the set of arcs emanating from $v$.
Specifically, the arc-type is the partition of $d$ as the sum
$$n_1 + n_2 + \dots + n_t + (m_1 + m_1) + (m_2 + m_2) + \dots + (m_s + m_s),$$
where $n_1, n_2, \dots, n_t$ are the lengths of the self-paired orbits, and
$m_1,m_1, m_2,$ $m_2, \dots, m_s,m_s$ are the lengths of the non-self-paired
orbits. %, in descending order.
This is a graph invariant.
For example, if $X$ is arc-transitive then its arc-type is $d$, while if
$X$ is half-arc-transitive then its arc-type is $(d/2 + d/2)$, and if $X$ is
zero-symmetric (or equivalently a graphical regular representation of the group $A$),
then $n_i = 1$ and $m_j = 1$ for all $i$ and $j$.
In this talk I will explain how it can be shown that every partition
of the given form occurs as the arc-type of some vertex-transitive graph,
with the exception of $1+1$ and $(1+1)$. The proof uses Cartesian products
of `relatively prime' examples of specially chosen VT graphs.
This is joint work with Toma\v{z} Pisanski and Arjana \v{Z}itnik (Ljubljana).

### Primary author

Prof.
Marston Conder
(University of Auckland)