Question 1 For nЄ N, let Gn be the set of all simple graphs G with V(G)= [n] and |E(G)| = n = 1. (e) Does there exist an element of G7 that is connected but not a tree? Justify your answer.

True or False. Justify Your Answer.  a) There is a vertex v in a bipartite graph G = (V,E) with deg (v) = m+n. Where m = |V1| and n = |V2| b) is it possible for a graph to be a tree and complete bipartite graph at the same time?

Can someone provide the proof for this? TIA

Let G be a graph of order n that is isomorphic to its complement G. How many edges does G have? Explain your answer. If a graph G has n vertices, all of which but one have odd degree, how many vertices of odd degree are there in G, the complement of G? Prove your answer.

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Consider the statement, For all graphs G, if G contains an odd cycle, then x(G) > 3. Note, you do not need to know anything about graphs to answer these questions. What would the first line of a direct proof be? A. Assume there is a graph G which contains an odd cycle but for which x(G) 3. O C. Let G be a graph and assume it contains an odd cycle. O D. Let G be a graph and assume x(G) 3. O E. Let G be a graph and assume it contains no odd cycles. What would the first line of a proof by contradiction be? A. Assume there is a graph G which contains an odd cycle but for which x(G) 3.