I am struggling with the 'homotopic' part of the question
Which of these are homotopic but not homeomorphic?
The number of vertices of degree $\neq 2$ is a topological invariant, thus is true for homeomorphic space.
Respective to the above picture the number of vertices of degree $1$ are: $2, 1, 0, 2, 3, 6, 4, 4, 0 $ and $1 $
Respective to the above picture the number of vertices of degree $3$ are: $ 2, 1, 2, 0, 1, 0, 0, 2, 0 $ and $1 $
This distinguishes all cases except for $\mathbb{\phi}$ and $\mathbb{6}$ which both have a degree $1$ vertex of degree $1$ and a degree $3$ vertex of degree $1$
These are the only ones that can be homeomorphic, but they are not since $\phi$ has a vertex of degree $4$ and $6$ does not.
So all graphs are not homeomorphic.
So I have done the homeomorphic part, but I am struggling on the homotopic part. Could you help me with that?