An inequality for the number of independent sets of matroids with an application to the forest-tree ratio of graphs

Abstract

Let $M=(E,\mathcal{I})$ be a matroid of rank $r$. Let $\mathcal{I}_k$ be the independent sets of size $k$, and let $I_k=|\mathcal{I}_k|$ and $I=|\mathcal{I}|$. We show that if every set $F\in \mathcal{I}_{r-1}$ is contained in at least $δ$ bases, then $$\ln \left(\frac{I}{I_r}\right)\geqslant \frac{I_{r-1}}{I_r}\cdot δ\ln \left(1+\frac{1}δ\right).$$ In particular, we have $$\frac{I}{I_r}\geqslant 2^{I_{r-1}/I_r}.$$ By combining this result with several other ideas, we prove that if $G$ is a simple connected graph on $n$ vertices, and $F(G)$ and $T(G)$ denote its numbers of spanning forests and spanning trees, respectively, then $$\frac{F(G)}{T(G)}\geqslant \frac{F(K_n)}{T(K_n)},$$ where $K_n$ is the complete graph on $n$ vertices. Equality holds if and only if $G=K_n$.