Upper bound on the number of edges given number of vertices, girth and maximum degree

Question

For a simple and undirected graph $G$, is there a known upper bound on the number of edges it has, given number of vertices $n$, girth $g$ and maximum degree $\Delta$?

Answer 1

No, there is no bound. For instance, consider an $n\times n$ square grid. It has girth $g=4$ and maximum degree $\Delta=4$ but there are roughly $2n(n+1)$ edges, which is unbounded as $n\to\infty$ (even though the girth and maximum degree are bounded.

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Question was edited to include number of vertices, but now there is a trivial bound: $e\leq n\Delta/2$, since the total number of edges is half the sum of the vertex degrees.

Answer 2

You should have a look at this survey, section 4, you should be able to find some results.

A first upper bound, without taking into account $\Delta$ would be, for odd girth, $$\textrm{ex}(n, \{C_3, C_4, \dots, C_{2k}\}) < \frac{1}{2} n^{1 + 1/k} + \frac{1}{2} n,$$

Where $\textrm{ex}(n,H)$ is the maximal number of edge in a graph not including $H$.