Let G be a simple graph with n vertices and m edges. Prove:
If n = 11 and m = 46, show that G is connected.
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.