Let $G$ be a weighted undirected graph with vertex set $V$ and weight matrix $W$. Given $S\subset V$, we define $$cut(S,S^c):=\sum_{i\in S,j\in S^c} w_{ij}$$ Let $L$ be the graph Laplacian of $G$. It is well known that the second smallest eigenvalue $\lambda_2(L)$ is linked to the existence of a of a cut in $V$; in particular, it has been proved that $$\text{min}_{S\subset V} \frac{cut(S,S^c)|V|}{|S^c||S|}\ge \lambda_2(L).$$ Do there exists any result that bounds this eigenvalue from the below?
$$\lambda_2(L)\ge ...$$ (even in some special cases)
The second smallest eigenvalue $\lambda_2$ is closely related to the isoperimetric constant of a graph. A general result is that if a graph $G$ has maximal degree $d$ and isoperimetric constant $iso(G)$, then we have $$iso(G) \leq \sqrt{2d\lambda_2}.$$
See Theorem 11.15, Bogdan Nica, A Brief Introduction to Spectral Graph Theory. The theorem is originally from this paper: https://www.ams.org/journals/tran/1984-284-02/S0002-9947-1984-0743744-X/S0002-9947-1984-0743744-X.pdf.