If $X$ is a locally finite graph, (i.e. each vertex has finite index), is it true that the automorphism group Aut($X$) of the graph X is locally compact?
Here, Aut($X$) has compact open topology; and topology of $X$ is the weak topology when we consider $X$ as a CW-complex (see. Hatcher, Algebraic Topology - Graphs and Free Groups).