Prove that a graph G is a forest if and only if every edge of G is a bridge. Prove that, if a tree T has a vertex of degree k $,$ then T has at least k end$-$vertices. Determine the number of vertices in the graph G with degree sequence $4,3,3,2,1,1.($a$)$ How many edges are there in G$?$ Explain how you obtain your answer. $($b$)$ Sketch the graph G with the above given degree sequence.