Graph theory possibilities

Question

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?

Answer 1

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 $\binom{n}{2}$

Answer 2

1) All cycles have $n$ vertices and $n$ edges.

2) No. A connected graph with $n$ vertices and $n-1$ edges is a tree. Therefore, it cannot have a circuit.

3) See $C_{3} \cup P_{3}$, where $C_{3}$ and $P_{3}$ are disjoint. So $C_{3}$ has three vertices and three edges, then $P_{3}$ has three vertices and two edges.