Keele Research Repository

We introduce notation and terminology to investigate conditions on a permutation group G sufficient to ensure that G fixes a graph in any switching class of graphs that it stabilises. We show that cyclic groups, groups of odd order groups of order ^k+2 and all stabilisers of switching classes of graphs on an odd number of vertices have this property.
In Chapter 5 we give a necessary and sufficient condition for a dihedral group to have this property.
In Chapter'6 we consider switching classes containing forests and graphs with a given girth g > 5. We give necessary and sufficient conditions for the stabilisers of all such switching classes to fix graphs in their classes.
Finally we give a brief account of the link between strong graphs and switching, and give an example of a class of switching classes with doubly transitive stabilisers.