I don't know a name, but some of these graphs come up in subfactor theory, as "principal graphs".

The first example is just the [Haagerup subfactor](https://arxiv.org/abs/0902.1294), with principal graph the $3$-long $3$-star in your terminology. Emily Peters has written a bit about this, and it has an [even stranger companion](https://arxiv.org/abs/0909.4099) where one leg is $7$-long. However, there's actually a subfactor with principal graph a $3$-long $p$-star for every prime $p$!