Chromatic Polynomial(Chromial) For a Tree

In graph theory, a chromatic polynomial is a great tool. It is used to study the coloring properties of graphs. Our primary focus in this article is discussing and deriving the chromatic polynomial for a tree.

Chromatic Polynomials

The chromatic polynomial is nothing but the number of possible ways one is able to color the vertices of a graph with some number of colors. If we have y colors and suppose there is a graph G, then it can be shown that f(G,y) is called the chromatic polynomial of G.(Here we have to ensure that adjacent vertices have different colors. The smallest value of y for which f(G,y) is not equal to zero is a chromatic number of the graph.

Chromatic Polynomial for a Tree

Tree 1.1

Let T be a tree with n vertices. Suppose we have λ colors to color the tree.

In the above picture(Tree 1.1), vertex, P is a pendent vertex and it is adjacent to only one vertex Q. So both can not be colored with the same number of colors. As P can not be assigned with the color of Q, so Q can be assigned with (λ-1) number of colors.

Continuing the same way vertex R can be assigned with (λ-1) number of colors. V also has a number of colors because it is adjacent to Q. Vertex U and S are also pendent vertices and are only adjacent to V so both can be assigned with (λ-1) number of colors.

By the principle of counting,

f(T,λ) = λ(λ-1)n-1 

Example :

Here we are going to use the same example from the above diagram.

Tree* 1.2

In the above diagram, we have used two colors Red and Blue to color the tree, enhancing clarity and comprehension.

According to the diagram (Tree 1.2) this tree has 6 vertices so the chromatic polynomial will be

f(T,λ) = λ(λ-1)6-1 = λ(λ-1)5

Conclusion

The chromatic polynomial, or chromial for a tree provides valuable insights into its coloring properties. It is the fundamental concept in graph theory and contributes to various fields like graphs, combinatorics, and optimization problems. We can also prove this by the method of induction.

Frequently Asked Questions

Q.1: What is tree in graph theory?

Answer:

It is a undirected , acyclic connected graph ,and there is a unique path between all pair of vertices.

Q.2: How can the chromatic polynomial be utilizedhave?

Answer:

It can be used to determine the minimum number of colors require to color any graph , it also known as chromatic number.