On Limits of Vertex-Symmetric Graphs and Their Automorphisms

Using a simple but rather general method of constructing Cayley graphs with trivial vertex stabilizers, we give an example of an infinite locally finite Cayley graph (and, hence, an example of an infinite connected locally finite vertex-symmetric unimodular graph) which is isolated in the space of connected locally finite vertex-symmetric graphs. We also give examples of Cayley graphs which are not isolated in this space but are isolated from the set of connected vertex-symmetric finite graphs.