# Maximal number of smallest circuits in a matroid

It is known (see here for example) that, in a simple graph of odd genus $$g$$ with $$n$$ vertices and $$m$$ edges, the number of cycles of lenght $$g$$ is at most $$\frac{n(m-n+1)}{g}$$.

Since this can be be phrased only in terms of circuits and matroid properties (when the graph is connected, we can recover $$n$$ as the rank of the matroid plus one), I was wondering is this result could be extended to any matroid. More precisey, is it true that, for any matroid of size $$m$$, rank $$r$$ and odd genus $$g$$, the number of circuits of size $$g$$ is at most $$\frac{(r+1)(m-r)}{g}$$?

If no, are there some weaker similar bounds on the number of circuits of size $$g$$?

If yes, what can we say about the equality case? In the case of graphs, it is reached for the Moore graphs. Are there new non-graphical matroids reaching the equality?

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• If you leave the world of graphical matroids, no such small bound can hold. The number of shortest circuits in a uniform matroid $U_{m,r}$ is $\binom{m}{r+1}$. – Gjergji Zaimi 2 days ago
• Oh, you're right... I was hoping that this bound could have given some matroid theoritic version of Moore graphs, but this probably can't work – Antoine Labelle 2 days ago