All nodes are color coded according to their number of divisors.
About the Goldbach conjecture prime graph
In this visualization if an even composite number is the sum of two primes,
say A and B, then there is a link, A --> B, between the two, and the weight
of that link is the composite number itself.
About the Goldbach conjecture composites graph
In this visualization if an even composite number C is the sum of two primes,
say A and B, then there we add the following links:
A --> C
B --> C
Both with the same weight, in this manner we create a mix of composite and prime
numbers, so the graph shows how "close" are they.
About the Number of divisors graph
In this visualization a link is made between a number and the quantity of
divisors it has, for example:
42 has 8 divisors: 1, 2, 3, 4, 6, 7, 21 and 42, so the link is 42 --> 8.
What makes this graph useful is being able to show the orbits of the divisor
function, that is, the path created by repeatedly applying the function. Like
so:
42 --> 8 --> 4 --> 3 --> 2
Ending in 2, because it has 2 divisors. In the above example the orbit has
length 4.
About the divisor function orbit graph
In this visualization the result from the orbits of the previous study is
depicted. Here nodes are linked in the following manner:
A --> B means that A has B steps to get to ne node 2, using the example from
the previous graph:
As the number 42 the path 42 --> 8 --> 4 --> 3 --> 2, then its orbit has
length 4, so the link 42 --> 4 is added.
About the prime gaps graph
In this visualization a link is made between the prime index and the difference
from the previous prime:
The second prime is 3, and the difference from the first prime, which is 2, is 1,
so the link is 2 --> 1 (the index, 2, links to the difference, 1).
This is a classic graph in the prime gaps subject, using the index instead of
the prime itself makes it connected.
Conclusions
This study shows that the two Goldbach based graphs are very different
for the divisor based ones, the latter having a structure that resembles a fractal.
The number of divisors property has shown to be a very strong grouping property, as
in second study the number of divisors where closely related to the orbit length.
Some number properties have easily been spotted in the two divisor based studies:
powers of numbers stayed close to the same nodes.
A surprising fact is the similarity between the prime gaps and the divisors graphs,
certainly this is a topic for further invetigation.
Further studies
Degree distribution
An important measure for graph classification is the distribution of degrees,
On the first sight, Goldbach based graphs have more distributed links, whereas
divisor function based graphs have nodes with concentrated links.
Mean degree change with number of nodes
Mean degree is the "central" indicator for the study of graph centrality, the
Goldbach based graphs show much lower centrality than the divisors based, studying
how the centrality changes when a new node is added is on topic for extending.
Orbit length
One interesting study is to model the length of the orbit with the number of nodes.
Relation between the divisors of composite numbers and the Goldbach primes
The second graph needs a more deep study to discover if there is a relation
between the number of divisors of a composite number and the 2 primes that
add to it.
The first question to be answered in this case is: which pair of primes to use?
As the number of ways of summing to an even composite number increases, which
pair should be considered, here the pair with the least prime was chosen.