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MATH 443 Problems #4 Due Tuesday October 22. These problems are for classroom presentations and some fraction of them may be final exam questions. 17. Let A be n × n the adjacency matrix of a graph G on n vertices where A = (aij ) and aij = 1 if ij ∈ E and aij = 0 if ij 6∈ E. Show that the i, j entry of Ak is the number of i-j-walks in G that use exactly k edges. 18. Assume diam(G) = k. Assume A is given as above. Show that the matrices I, A, A2 , . . . , Ak are linearly independent. 19. Let D = (N, A) be a tournament i.e. an orientation of the edges of a complete graph Kn (so |N | = n). Show that D has a directed spanning path. You should look for the longest path! 20. Assume D = (N, A) is strongly connected. Show that there is a directed spanning subgraph D0 = (N, A0 ) with A0 ⊆ A satisfying |A0 | ≤ 2|N | and D0 is strongly connected. 21. Assume G is connected. Show that κ(G) = 1 + min κ(G − v). v∈V (G) 22. A partially ordered set on a finite set of elements X gives comparisons between some but not necessarily all pairs of elements. We can think of this as a digraph D = (X, A) where we have the properties that i) If a → b then b 6→ a, ii) If a → b and b → c, then a → c. iii) D is acyclic (no directed cycles). Assume |X| ≥ rs + 1. Show that there is either a chain of r + 1 elements (r + 1 elements a1 → a2 → · · · → ar+1 ) or an antichain of s + 1 elements (s + 1 elements A = {a1 , a2 , . . . , as+1 } so that for ai , aj ∈ A we have ai 6→ aj and aj 6→ ai ).