By definition, an $n$-star graph, $n \geq 3$, consists of $n+1$ nodes: one node of degree $n$ and $n$ leaves. This is just the complete bipartite graph $K\_{1,n}$.

Is there a name for a star-like graph where instead of leaves we allow paths of arbitrary length? 
In other words, a tree with exactly one node of degree $n$ and all other nodes of degree $\leq 2$.