A name for star-graph with long "laces"

Question

An $l$ long $k$-star is a graph with centeral vertex $o$ which is connected to $k$ line graphs of length $l$.

For example a 2-long 3-star looks like:

x1-x1-O-x2-x2
      |
      x3-x3

$o$ is the central node, $x1,x2,x3$ are the three line-graphs connected to $o$.

Rigorously speaking, an $\ell$-long $k$ star is composed of:

1. $k$ line graphs $l_1,l_2,\dots,l_k$ each with $\ell$ vertices
2. centeral vertex $o$
3. $k$ edges $(o,l_1),(o,l_2),\dots,(o,l_k)$

Is there a "standard" name for such a graph family?

Answer 1

It is sometimes referred to as "spider graph".

Answer 2

The usual term for such a graph is "star-shaped," though that usually doesn't require that the arms be of equal length. Star-shaped graphs are important in the solution of Deligne-Simpson problem, which is to understand the space of ordered n-tuples of matrices with fixed eigenvalues that multiply to 1. See the papers of Bill Crawley-Boevey (arXiv, MathSciNet).

Answer 3

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 $p$!