Have you ever wanted to take the inverse of a function but you have to take a determinant of a n by n matrix where n is the number of vertices of a graph?
Of course you have!
Do you want familes of complete graphs to be (a-1)^blah(a+n-1)(1-u)^r-1 or something. Well, you can ! You can thank James Joesph Sylvester because of their determinant identity. You can also thank them for a lot of things, they were quite clever.
As a sidenote: They are the reason we say matrix, graph, discriminant, and totient. Very cool stuff.
Extra sidenote: they were denied a degree for a while for being jewish, not very cool stuff.
Pretty much all the simplification that we did was massaging matrices to work with Sylvester determinant identity to create polynomials.
I don't actually know a lot about Zeta functions, Graph theory, etc...
