Question 1 For n € N, let Gn be the set of all simple graphs G with V(G)= [n] and [E(G)| = n − 1. (a) Draw all elements of G3, i.e., all simple graphs with vertex set {1,2,3} and 2 edges.

For which of the following does there exist a simple graph G = - (V, E) satisfying the specified conditions? Select one: O A. It has 7 vertices, 10 edges, and more than two components. B. It has 8 vertices, 8 edges, and no cycles. O C. It has 6 vertices, 11 edges, and more than one component. O D. It is connected and has 10 edges, 5 vertices and fewer than 6 cycles. O E. It has 3 components, 20 vertices and 16 edges.

Draw a Cayley graph for Z/7Z with generating set {2¯,5¯}.

Let G (V, E) be a graph such that = • V ≥ 3, G has exactly 2 leaves, and in G all the non-leaf vertices have degree 3 or more. Prove that G has at least one cycle. You are not required to draw anything in your proof.