Q: A graph is a pair X = (V, E) where V is a set of elements that we call vertices and E is a set of 2-element subsets of V that we call edges......

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4. A graph is a pair X = (V, E) where V is a set of elements that we call vertices and E is a set of 2-element subsets of V that we call edges. For instance, the figure below shows a graph with vertex set V = {1, 2, 3, 4} and edge set E = {12, 23}. 2 3 Let Sy be the set of permutations of V. A map o E Sy is called a graph automorphism if for every uv E E we have o(u)o(v) E E. We denote by Aut (X) the set of all graph automorphisms of X. For instance, in the example above the permuation o given by o(1) = 3, o(2) = 2, 0(3) = 1 and o(4) = 4 is a graph automorphism. (a) Show that for every graph X, the set Aut(X) is a group under composition. In particular notice that Aut(X) acts on V. (b) Let X = (V, E) be the 5-cycle, that is V = Z; and ij E E if and only if j = itl mod 5. Find Aut(X), prove all your claims. (c) Let X = (V, E) be the n-clique, that is V = {1, ...,n} and ij E E for every pair of vertices i, j E V. Find Aut(X), prove all your claims