4. (a) If a graph has 31 edges, what is the smallest number of vertices it can have? (b) If a connected planar graph with n vertices, all of degree 4 has 10 regions, determine n is a 12-vertex planar graph with e edges, determine all possible values of e. Sketcha of vertices that a bipartite planar graph with 16 edges can Sketch an example of such a connected planar graph 12-vertex planar graph with e edges, where e is a maximum. (d) What is the smallest number 2 Give an example of a bipartite planar graph attaining this number of vertices. (4)