Question on definition of group acting on a topological space. I know that a group can act on a graph by acting on the set of vertices on a graph. I also know that a graph can be viewed as a CW complex and therefore a topological space and i am trying to bridge the two together through defining group actions on graphs topologically.
I am however struggling with the definition of group actions on topological spaces. 
The definition reads that a group action $G$ on a topological space $X$ is a group action of a topological group G that is continuous i.e.
$G \times X \longrightarrow X$ is a continuous map
How would this work for a group acting on a graph, surely mapping the vertices of a graph isn't continuous. 
Any help on this would be greatly appreciated.
 A: When a group acts on a graph (in the usual sense), there is more structure than just a permutation of the vertices. Specifically, it not only maps vertices to vertices, but preserves the property that if there is an edge $(a,b)$ in the graph, then there will be an edge $(g(a), g(b))$ as well. That is, it permutes both the vertices and edges in a compatible way, or if you prefer, each element of the group is a graph isomorphism.
This aligns exactly with the action of a group on Hatcher's topological definition of a graph. The ``graph'' is a bunch of points representing vertices, with line segments glued between them to represent edges. A group acting on the graph (in the usual sense) could be realized by a group of continuous functions that permute the vertices, and map the points of the edge $(a,b)$ homeomorphically to the points of the edge $(g(a), g(b))$.
It is also possible for other groups of homeomorphisms to act on Hatcher's construction of a graph, though. For example, the topological version of the graph $K_3$ is homeomorphic to a circle, so there are infinitely many homeomorphisms of that space, and they form a group. By comparison, the group of graph isomorphisms of $K_3$ has size 6.
