Construct a graph satisfying the following properties. (23p) 1. It must have 6 vertices. 2. The set of vertices must be V={a,b,c,d,e,f}. 3. There must not be any two and more edges in the same direction between any two vertices. 4. It must have at least three loops. Which vertices are they? 5. At least four distinct vertices are both initial point and terminal point except the case to be a loop. Which vertices are they? 6. It must include at least four edges with bidirectional. Which vertices are they? 7. It must not have an isolated vertex. 8. It must include a circuit. Which vertices are making up that circuit ? 9. It must include (at least one) a sequence of vertices which is not path. Give an example. Answer the following questions for the graph you have constructed. Q1. Answer the questions in the items above with respect to your graph. Q2. Determine in-degree of all the vertices, respectively. (12p) Q3. Determine out-degree of all the vertices, respectively. (12p) Q4. Write the relation associated with the graph you have constructed by using the set notation. (12p) Q5. Check if your relation is reflexive, symmetric, antisymmetric, transitive. Explain your answer in detail. (8P) Q6. What are the definitions of the partial ordering and the equivalence relation? Is your relation a partial ordering? Is your relation an equivalence relation? If your answer is yes, why? If your answer is no, why not? (8p) Q7. Write the adjacency matrix which represents your graph. (12p) Q8. Construct an adjacency list for your directed graph. (8p) Q9. Is "cbed f' a path in your directed graph? If your answer is yes, why? If your answer is no, why not? (5p)