The question given to me is:
Let $G$ be a regular graph with $22$ edges, how many vertices can $G$ have?
I'm trying to figure out how to generalise my answer to this to all possible values of V. So from the question given to me I have $E = 22$ where $E$ is number of edges.
Using Euler's formula ($V-E+F = 2$) i know $V+F=24$, here we can work our way through one at a time through each of the 23 potential vertices to see if they are regular. E.g if $F = 1$ then $V = 23$ however this is not regular as all vertices have degree $2$ except two which have degree of $1$. Then move onto the next and see that when $V = 22$ we do have a regular graph and so on and so forth. However how do I go about generalising this into a viable proof. As my method is very long winded.