Suppose $G$ is an undirected $d$-regular $n$-vertex graph for some constant $d$. Let $\lambda_k$ be the $k$-th largest eigenvalue of the normalized laplacian $L$ of $G$ (defined as $I - \frac{1}{d} A$ where $A$ is the adjacency matrix of $G$). If the degree is constant, we can have good expanders of course but I want to say that the lower eigenvalues cannot be too close to $1$ if the degree is constant. I was wondering if there is a quantitative statement of this sort. Are there any bounds known about how close $\lambda_k$ can be to $1$, as a function of $k$? In particular, is something of the following form known:``For all $ k \ll n$,  we have $\lambda_k  \; \leq \; \left( 1 - O\left( \frac{1}{k^{0.99}}\right) \right)$'' ?(where the constant in the order notation can hide factors of $d$).
Upper bounds on higher order eigenvalues of regular graphs
spectral graph theory