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.