Is there an algorithm to generate graph edges given amount of vertices and edges per node? [closed]

Question

Is there currently an algorithm that given the number of vertices / nodes n, and the number of edges per node l, output all graph edges?

If not, is there an algorithm that can tell if such graph is possible?

In both cases, this is an undirected, loop-free graph.

Answer 1

From the comments, you want every node to have the same weight. This is of course only possible if $n \geq \ell + 1$. Now assume we have such parameters. Pick $\ell+1$ vertices and form a complete graph on them. Continue, until you only have $k \leq \ell$ vertices left. Now there are two cases:

1. $\ell - k$ is odd. Then form a complete graph on the remaining $k$ vertices, now every remaining is still missing $\ell - k-1$ edges, an even number. Remove an edge $(v,w)$ from the previous graphs and connect a remaining vertex to both $v$ and $w$, until everyone has the desired degree. You will have to remove $$\frac{k(\ell - k)}{2}$$ edges, which is possible, as each complete graph generated before contains $$\frac{\ell(\ell+1)}{2}$$ edges.
2. $\ell - k$ is even. Then do not form the complete graph but rather connect every vertex directly to $\ell$ vertices by removing edges. Once again, there are enough edges to remove. This only works if $\ell$ is even, but we can assume $\ell$ even due to the handshaking lemma, which states that if $\ell$ and $n$ are both odd, then such a graph does not exist.