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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 subfactorHaagerup subfactor, 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 companioneven stranger companion where one leg is $7$-long. However, there's actually a subfactor with principal graph a $3$-long $p$-star for every prime $p$!

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, 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 where one leg is $7$-long. However, there's actually a subfactor with principal graph a $3$-long $p$-star for every prime $p$!

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, 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 where one leg is $7$-long. However, there's actually a subfactor with principal graph a $3$-long $p$-star for every prime $p$!

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Kim Morrison
Kim Morrison
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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, 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 where one leg is $7$-long. However, there's actually a subfactor with principal graph a $3$-long $p$-star for every prime $p$!