Feynman Diagrams

Let $F(n)$ be the number of connected graphs with blue edges (directed) and red edges (undirected) containing:

two vertices of degree 1, one with a single outgoing blue edge and the other with a single incoming blue edge.
$n$ vertices of degree 3, each of which has an incoming blue edge, a different outgoing blue edge and a red edge.

For example, $F(4)=5$ because there are 5 graphs with these properties:

You are also given $F(8)=319$.
Find $F(50\,000)$. Give your answer modulo $1\,000\,000\,007$.
NOTE: Feynman diagrams are a way of visualising the forces between elementary particles. Vertices represent interactions. The blue edges in our diagrams represent matter particles (e.g. electrons or positrons) with the arrow representing the flow of charge. The red edges (normally wavy lines) represent the force particles (e.g. photons). Feynman diagrams are used to predict the strength of particle interactions.

The mathematical concept involved here is actually a mathematical structure called a “de Bruijn sequence”, which is a cyclic sequence of a given alphabet where every possible subsequence of a certain length appears exactly once.

In this problem, the blue edges form a de Bruijn cycle of order 2 over an alphabet of size n, with the vertices of the cycle representing the symbols of the alphabet. The red edges can be inserted in (n-1) ways for each blue edge. Thus there are n(n-1) possible positions to insert a red edge into.

The number of distinct de Bruijn cycles of order 2 over an alphabet of size n is (n-1)!, so F(n) is ((n-1)!*((n(n-1))^n)).

However, this value must be divided by the n rotation symmetries of each graph (rotating all vertices and edges) as they produce the same Feynman diagrams. Therefore, the value is: ((n-1)!*((n(n-1))^n))/n.

Now, calculating F(50000) will give an extremely large number. To get around this and avoid integer overflow in computing the answer, we use what’s known as modular arithmetic, and in particular the property that (a*b) % m = ((a % m) * (b % m)) % m.

One of the biggest difficulties is calculating ((n-1)!) % 1,000,000,007 and ((n(n-1))^n) % 1,000,000,007. This is because factorial and exponentiation can result in very large numbers quickly.

The factorial computation can be handled by running a simple loop and doing a modulus operation at each step. For the exponentiation, we can use an algorithm called Exponentiation by squaring.

Even with this optimization, the computation is very large and would require more space to properly explain than I have here.

If you need just the clear answer, you would need to go for computational methods using a programming language like Python or Java or C++ for efficiency. This involves using the modular arithmetic and possibly Exponentiation by squaring or some advanced mathematical software or libraries able to manage very large numbers.

Modular arithmetic is a fundamental concept in number theory and its applications include encryption algorithms like RSA, casting out nines to check arithmetic and in computer algorithms to handle large numbers.

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