Mathematics 4. Color polynomials of Graphs

Today (), we talked about the color polynomials of graphs. It is a good exercise to prove that the result is a polynomial.

The most common and simplest relationship in mathematics is linear relationship, which is the content of linear algebra learning.

Then it is a polynomial relationship. Many finite-element problems can produce polynomial expressions. This is a classic example.

 

A graph \ (g \) is a finite set. It contains a finite vertex and a finite edge.

One color of the graph \ (g \) is to dye the \ (g \) vertex. Requirement: if the two vertices are connected by edges, the two dots are colored differently.

Given the colors \ (g \) and \ (k \), the total number of dyeing methods is counted as \ (p_g (k )\). proof: \ (p_g (k) \) is a polynomial of \ (k. That is, the polynomial \ (f (k) \) causes \ (p_g (K) = f (k )\).

 

Tip:

1. Calculate a few simple graphs first.

2. for a given graph \ (g \), subtract an edge \ (E \) and record it as \ (G '= g-e \), deduce \ (p_g (k )\) and \ (P _ {G'} (k.