On the Turán number of double stars

The Turán number of a graph $F$, $ex(n,F)$, is the maximum number of edges in a graph on $n$ vertices which does not contain $F$ as a subgraph. Let $S_{a,b}$ denote a double star with a central edge $uv$, $a$ leaves connected to $u$ and $b$ leaves connected to $v$. The function $ex(n,S_{a,b})$ has been studied for $a=1,2$, their extremal graphs are disjoint copies of $K_{a+b+1}$ and either a small clique or a near $b$-regular graph. In this paper, we further study $ex(n,S_{3,b})$ and determine the extremal graphs, which have more structures than those of $a=1,2$.