1) (3pts) Let G be a connected graph. Prove that G is Eulerian if and only if each of the blocks of G is Eulerian. 2) (2.5) A wheel graph W,, n 2 4 is the graph made from the cycle C,-, by adding an additional vertex v, (in the center of the cycle) and additional edges connecting Un to each vertex of C-1- Describe the graph W, and prove that W, is Hamiltonian. 3) (3) Let D be a digraph with nodes V = (a, b, c, d, e) and arcs E = ((a, b), (a, c), (a, d), (b, e), (c, d), (d, a), (d, e), (e, b)). Find the strongly connected components of D. Find the directed cycle of largest order of D and find the cycle of largest order in the underlying graph of D. 4) (3) Consider the complete bipartite graph K2 3 on partite sets (a, b], (c, d, e). Give an orientation D of K23 where D is strongly connected. Prove that D is strongly connected. 5) (3) Let G be a graph. Prove that G is Eulerian if and only if G has an orientation D where D is an Eulerian digraph. 6) (3) For the following digraph D, find the incidence matrix B using the node set V = {a, b, c, d, e) and the arc set E = (1,2,3,4,5,6). Compute BBT and find its relationship to the degree matrix and the adjacenty matrix of the underlying graph of D. 7) (2.5) Describe the arc set of a tournament D of the complete graph Ks on the nodes V = {a, b, c, d, e) so that D contains a Hamiltonian path but D is not a Hamiltonian digraph. List a Hamiltonian path