4. For each exercise below, assume the graph is a simple undirected graph. (a) Give the formula for the number of edges in a tree, denoted ITl, as a function of the number of nodes, n, that the tree has. [5 points] (b) A forest is an undirected graph that consists of connected components that are trees. Suppose a forest consists of k connected components and n nodes. Give a formula for the number of edges that the forest has, denoted |Fll, as a function of n and k. [5 points] (c) Draw a 3-regular graph on six vertices. [5 points] DEFINITION 1. An r-regular graph is a graph whose vertices are r degree each. (d) Professor Tout Savoir claims that every complete graph on n vertices is a regular graph. If her claim is true, give an expression for r in terms of n for which a complete graph on n vertices can be described as an r-regular graph. On the other hand, if her claim is false, give a counter-example to disprove it. [5 points] (e) Professor Tout Savior also claims that for any pair of nonnegative integers n and r, n > r, one can construct an r-regular graph on n vertices. If her claim is true, explain why. If her claim is false, for what values of n and r will it be false. [5 points]