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. (a) Draw a connected Eulerian graph that has exactly four vertices and five arcs but is not simple.
   (b) Explain why it is not possible to have a simply connected Eulerian graph with exactly four vertices and five arcs. A simply connected Eulerian graph is drawn that has exactly eight vertices and ten arcs.
2. (a) Explain how you know that the sum of the vertex orders must be 20 .
   (b) Write down the minimum and maximum possible vertex order and draw a diagram that includes both the minimum and the maximum cases.
   (c) Draw a diagram to show a simply connected Eulerian graph with exactly eight vertices and ten arcs in which the number of vertices of order 4 is as large as possible.