9. One of the basic motivations behind the Minimum Spanning Tree Problem is the goal of designing a spanning network for a set of nodes with minimum total cost Here we explore another type of objective designing a spanning network for which the most expensive edge is as cheap as possible. Specifically, let (V.) be a connected graph with vertices, edges, and positive edge costs that you may assume are all distinct. Let T-(V.E) be a spanning tree of G; we define the bottleneck edge of T to be the edge of Twith the greatest cost. A spanning tree T of G is a minimumhorneck ng tree if there is no spanning tree To G with a cheaper bottleneck edge. (a) is every minimum-bottleneck tree of G a minimum spanning tree of G? Prove or give a counterexample (b) is every minimum spanning tree of G a minimum-bottleneck tree of G? Prove or give a counterexample. 9. One of the basic motivations behind the Minimum Spanning Tree Problem is the goal of designing a spanning network for a set of nodes with minimum total cost Here we explore another type of objective designing a spanning network for which the most expensive edge is as cheap as possible. Specifically, let (V.) be a connected graph with vertices, edges, and positive edge costs that you may assume are all distinct. Let T-(V.E) be a spanning tree of G; we define the bottleneck edge of T to be the edge of Twith the greatest cost. A spanning tree T of G is a minimumhorneck ng tree if there is no spanning tree To G with a cheaper bottleneck edge. (a) is every minimum-bottleneck tree of G a minimum spanning tree of G? Prove or give a counterexample (b) is every minimum spanning tree of G a minimum-bottleneck tree of G? Prove or give a counterexample