A connected graph is a graph with no disjoint subgraphs. A simple graph is a graph with no loops or multiple edges.
Easy Question: What are the necessary and sufficient conditions on the order (number of vertices) and size (number of edges) of a given graph in order to ensure that it is simple and connected?
For example, the only simple and connected graph of order $2$ and size $1$ is the path graph $P_2$ (a line segment). There is no simple and connected graph of order $2$ and size greater than or equal to $2$. A graph with order and size both $3$ is the cycle graph of order $3$ (a triangle). There is no simple and connected graph of order $3$ and size greater than or equal to $4$. Continuing in this fashion, I conclude that there must be a condition on the difference between the size and order (compared to the size).
Harder Question: For any pair of positive integers $(a,b)$ is there a simple and connected graph $G_{ab} = (V_{ab}, E_{ab})$ with order $|V_{ab}| = \frac{1}{2}(ab + a + b + \text{gcd}(a,b))$ and size $|E_{ab}| = ab$? If so, are such graphs significant?
For instance, if $b = 1$, then $|V_{ab}| = a + 1$, $|E_{ab}| = a$ and $G_{ab}$ can be realized as the path graph $P_{a+1}$ (or any tree with the same order), which is both simple and connected.