50.7k views
5 votes
Graphs from the form k 1,n are called stars prove that if kr,s is a tree then it must be a star

1 Answer

1 vote

Final answer:

A K_{1,n} graph is a star in which one central vertex is connected to n other vertices. To prove any tree K_{r,s} must be a star, understand that a tree is a connected graph with no cycles. K_{r,s} can only be a cycle-free tree if r or s equals 1, hence it is a star.

Step-by-step explanation:

The student question relates to graph theory, which is a part of discrete mathematics. A K_{1,n} graph, commonly known as a star graph, is a type of graph where one central vertex is connected to n other vertices, and those n vertices are not connected to each other. To prove that any tree K_{r,s} is a star graph, we first need to understand the definition of a tree.

A tree is a connected graph with no cycles. This means there is exactly one path between any two vertices. Suppose our K_{r,s} graph is a tree, it cannot contain a cycle of vertices. Now, if either r or s is equal to 1, it is trivially a star. However, if r and s are both greater than 1, this would imply multiple paths between some vertices, which contradicts the definition of a tree. Therefore, for K_{r,s} to be a tree, we must have one of r or s equal to 1, making it a star graph, K_{1,n}.

This provides us insight into the structure of trees and their relationship with star graphs within the field of graph theory.

User Berndh
by
7.6k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories