1. Graph theory help!

Let G be a simple graph with n vertices and m edges. Prove:

If n = 11 and m = 46, show that G is connected.

2. Originally Posted by walreinlim88
Let G be a simple graph with n vertices and m edges. Prove:

If n = 11 and m = 46, show that G is connected.
between every two vertices we can have at most 2 edges. so the maximum number of edges is $\displaystyle \binom{n}{2}=\frac{n(n-1)}{2}$.

EDIT: my argument for the second part was not complete. see the next post.

3. for the second part of your problem i'll prove a general result:

Claim: suppose a simple graph has $\displaystyle n$ vertices and $\displaystyle m$ edges. if the graph is not connected, then $\displaystyle m \leq \binom{n-1}{2}.$

Proof. so the graph has a connected component with $\displaystyle 1 \leq r \leq n-1$ vertices which is disconnected from the rest of the graph, which has $\displaystyle n-r$ vertices. therefore $\displaystyle m \leq \binom{r}{2} + \binom{n-r}{2}.$

it's now easy to see that the inequality $\displaystyle \binom{r}{2} + \binom{n-r}{2} \leq \binom{n-1}{2}$ is equivalent to $\displaystyle (r-1)(n-r-1) \geq 0,$ which is a true statement because $\displaystyle 1 \leq r \leq n-1.$ Q.E.D.