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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?

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4 Answers 4

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.

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It is sometimes referred to as "spider graph".

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Yes. Example: graphbestiarium.blogspot.com/2009/03/spiders.html –  David Eppstein Jan 4 '10 at 17:06

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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2  
I think that's not true ... though Izumi's paper (structure of sectors II) is a little hard to parse, Noah and I recently agreed that he asks/conjectures the existence of the 3-long p-star, but only constructs it for p=3 and p=5 –  Emily Peters Nov 1 '09 at 22:04

Is it not just a subdivision of a star graph?

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The family described above is more specific, since many subdivisions of star graphs don't have all arms the same length. –  Gabe Cunningham Jan 4 '10 at 22:01

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