Question 6 - A-Level Maths

6 The complete graph \(K _ { n } ( n > 1 )\) has every one of its \(n\) vertices connected to each of the other vertices by a single edge.

Draw the complete graph \(K _ { 4 }\).
1. Find the total number of edges for the graph \(K _ { 8 }\).
2. Give a reason why \(K _ { 8 }\) is not Eulerian.
For the graph \(K _ { n }\), state in terms of \(n\) :
1. the total number of edges;
2. the number of edges in a minimum spanning tree;
3. the condition for \(K _ { n }\) to be Eulerian;
4. the condition for the number of edges of a Hamiltonian cycle to be equal to the number of edges of an Eulerian cycle.