Prove that there exists a maximum flow f in G, such that the set of edges carrying positive flow is acyclic. In other words, f has the property that for every directed cycle in G made up of edges e1, e2, . . . , ek, at least one of the values f (e1), f (e2), . . . , f (ek) is equal to zero.

Give a condition that is necessary but not sufficient for an undirected graph to have an Eulerian Path. Justify your answer.

Consider the following pair of adjacency matrices.1. Draw the simple graphs associated with each of the above adjacency matrices.2. Check whether those two simple graphs are isomorphic. Show your work. 1  0  1                                 1  1  0  0  1  1                                 1  0  1  1  1  0                                 0  1  1

Assume that the Hamiltonian cycle problem for undirected graphs is NP-complete. Show that the Hamiltonian cycle problem is NP-complete for directed graphs. Note: Just need to reduce the Hamiltonian cycle problem for an undirected graph to a directed graph: for any edge (u, v) in the undirected graph, add two directed edges u->v and v->u.