2 A simple graph is one in which any two vertices are directly connected by at most one arc and no vertex is directly connected to itself. A connected graph is one in which every vertex is joined, directly or indirectly, to every other vertex. A simply connected graph is one that is both simple and connected.

1. What is the minimum number of arcs that a simply connected graph with six vertices can have? Draw an example of such a graph.
2. What is the maximum number of arcs that a simply connected graph with six vertices can have? Draw an example of such a graph.
3. What is the maximum number of arcs that a simply connected Eulerian graph with six vertices can have? Explain your reasoning.
4. State how you know that the graph below is semi-Eulerian and write down a semi-Eulerian trail for the graph.
   \includegraphics[max width=\textwidth, alt={}, center]{f0ac479b-2187-4335-bcb9-c0e354145bca-2_654_887_1457_587}
   \includegraphics[max width=\textwidth, alt={}, center]{f0ac479b-2187-4335-bcb9-c0e354145bca-3_549_1360_324_347}
5. Apply Dijkstra's algorithm to the copy of this network in the answer booklet to find the least weight path from \(A\) to \(F\). State the route of the path and give its weight. In the remainder of this question, any least weight paths required may be found without using a formal algorithm.
6. Apply the route inspection algorithm, showing all your working, to find the weight of the least weight closed route that uses every arc at least once.
7. Find the weight of the least weight route that uses every arc at least once, starting at \(A\) and ending at \(F\). Explain how you reached your answer.