3 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. Explain why it is impossible to draw a graph with exactly five vertices of orders \(1,2,3,4\) and 5.
2. Explain why there is no simply connected graph with exactly five vertices of orders \(2,2,3,4\) and 5 . State which of the properties 'simple' and 'connected' cannot be achieved.
3. Calculate the number of arcs in a simply connected graph with exactly five vertices of orders 1 , 1, 2, 2 and 4 . Hence explain why such a graph cannot be a tree.
4. Draw a simply connected semi-Eulerian graph with exactly five vertices that is also a tree. By considering the orders of the vertices, explain why it is impossible to draw a simply connected Eulerian graph with exactly five vertices that is also a tree. In the graph below the vertices represent buildings and the arcs represent pathways between those buildings.
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5. By considering the orders of the vertices, explain why it is impossible to walk along these pathways in a continuous route that uses every arc once and only once. Write down the minimum number of arcs that would need to be travelled twice to walk in a continuous route that uses every arc at least once.