how do I calculate the ismorphism group of a six-nodes-tree? How do I calculate the ismorphism group of a connected six-nodes-tree? The tree has a node centred and the other 5 nodes are leaves of the graph. I already know the answer is 6, which is the quotient between 720/120. But I do not know where 120 comes from. 
 A: If I understand your description correctly, you have the star graph $S_5$ with one central vertex and $5$ leaves radiating from it. Any automorphism of $S_5$ must send the central vertex to itself, because that’s the only vertex of degree $5$, and automorphims preserve vertex degrees. However, it can permute the other five vertices arbitrarily, so it is in fact $\operatorname{Sym}(5)$, the symmetric group on $5$ objects. There are $5!=120$ permutations of $5$ objects, so the order of this group is $120$.
A: You appear to be seeking the number of labelled graphs isomorphic to the star graph $K_{1,5}$, i.e., the size of its isomorphism class.

In general, the size of the isomorphism class a graph $G$ belongs to is given by $$\frac{|V(G)|!}{|\mathrm{Aut}(G)|},$$ where $V(G)$ is the set of vertices of the graph, and $\mathrm{Aut}(G)$ is the automorphism group of the graph.
For $K_{1,5}$, the automorphism group has size $5!=120$: we can permute the $5$ degree-one vertices arbitrarily and the degree-5 vertex must be fixed.  Hence the size of the isomorphism class is $$\frac{6!}{5!}=6.$$
The six labelled graphs in this isomorphism class are:

(Recall that two labelled graphs on the same vertex set are equal if they have the same edge set.)
