(3) Let n be a positive integer. Define a simple graph Qn as follows: The vertices are n-tuples (x1,... , xn) with x; E 10, 1) . There is an edge between (xi,... ,xn) and (y,... , yn) if they agree in exactly n-1 coordinates (i.e., there exists i such that aj Here are drawings of Q2 and Q3: Uj ifj 2 but x, y.) 001 011 101- 111 01 100- 110 10 010 a) Prove that Qn is connected. (b) How many vertices does Qn have? How many edges? (c) For what values of n does Qn have a closed Eulerian trail? (d) Prove that if n 2 2, then Q has a Hamiltonian cycle. (e) Let be a permutation of [n]. Show that the function (xi, . . . , Xn)- ((1), . . . , zo(n)) is an automorphism of Qn. Give an example (for all n) of another automorphism of Qn which is not of this form