An automorphism of a graph $G = (V,E)$ is a permutation $\sigma$ of the vertex set $V$, such that the pair of vertices $(u,v)$ form an edge if and only if the pair $(\sigma(u),\sigma(v))$ also form an edge.
On the other hand we can view a graph as a cell complex with usual topology. Let us denote it by $||G||$. Since $||G||$ is a topological space, we can talk about homeomorphism group of $||G||$. Recall that a function $f\colon ||G||\to ||G||$ is a homeomorphism if it has the following properties:
- $f$ is a bijection.
- $f$ is continuous.
- $f^{-1}$ is also continuous
Now I am wondering under which condition we can embed $Aut(G)$ into $Homeo(||G||)$.