Is it possible to have a simple graph(no loops or parallel edges), connected, six vertices, six edges?
Is it possible to have a graph, connected, ten vertices, nine edges, nontrivial circuit?
Is it possible to have a graph, six vertices, five edges, not a tree?
Best Answer
For the first consider a hexagon.
For the second consider a path with ten vertices.
For the third consider a pentagon and an isolated vertex.
Edit: The number of graphs with $n$ vertices and $k$ edges (If the vertices have allready been labelled) is $\binom{\binom{n}{2}}{k}$. This number is positive for values of $k$ between $0$ and $\binom{n}{2}$