What is the name of a graph that has $1$ central node connected to $n$ other nodes, each of them connected to $n-1$ distinct nodes, and so on?
At the end of the process the central node has degree $n$, as do all the other nodes apart from the last layer that have degree 1 (the leaves). I would like to know if this kind of graph has a specific name, and if there are known properties.
In this picture we see n=3 l=2 how it will appear a tree for n = 3 at the second layer, the zero layer is the root, the first layer are the 3 nodes connected (n) the second layer are the n-1 new nodes connected with the parents node.
I hope it will help
Best Answer
The infinite version of this graph -- so continuing indefinitely, no layer of vertices of degree one -- is called the regular tree (or Bethe lattice) of degree (or coordination number) $n$ in maths and statistical mechanics. It has its own wikipedia page. Inspired by the definition of 'perfect binary tree' on wikipedia a suggestion could be the regular tree (or Bethe lattice) of a given depth? For related terminology see Wikipedia on Trees.