# Breaking Graph-drawing Software

This, apparently, is what happens when you try to compile a graph with 16,410 edges using GraphViz’s dot:

$n=300$, dot, multigraph

It’s hard to tell what’s going on here; perhaps it would be a little easier on a much smaller subset of the graph; with 1531 edges, rendered under fdp:

$n=100$, fdp, multigraph

A bit clearer, but still incredibly messy.

$n=50$, fdp, multigraph

Figured it out yet? Well, it’s pretty obvious that the vertices are numbers — and specifically prime numbers. But it’s also a multigraph; that’s interesting, but we could do without the noise, so here’s a version with all redundant edges removed (dot also works better than fdp here):

$n=50$, dot, with no duplicate edges

Okay, so I’ll cut to the chase: they’re graphs, $G_n=(V_n, E_n)$, where $V_n$ is the set of prime numbers under $n$ and two primes $\pi, \pi' \in V_n$ have an edge connecting them if there exist two weak Goldbach summations $\pi = p_1 + p_2 + p_3, \pi' = p_1' + p_2' + p_3'$ (where $p_1, p_2, p_3, p_1', p_2', p_3'$ are all prime) that agree on two terms. I don’t know if these graphs have any mathematical significance whatsoever, but nonetheless I found it pretty interesting to think about. In particular, I’m wondering if inferring a simplicial complex from the complete subgraphs on these might render some insights; I notice $K_6$ is induced by $11,13,17,19,23,$ and $29$; I have to wonder how and in what way these graphs grow as additional primes are added.

$n=30$, dot, no redundant edges

$n=31$, dot, no duplicate vertices

$n=37$, dot, no duplicate edges

Also the $n=100$ case is slightly more intelligible without duplicate edges.

$n=100$, dot, no duplicate edges

(Mind you, I very well might’ve made errors in my code, and there may be some edges erroneously present or absent. It’s late and I’m too exhausted to think about that at the moment.)