讲座内容：

Let $G$ be an $n$-vertex graph with Laplacian eigenvalues $0=\lambda_1(G)\le \lambda_2(G)\le\cdots\le \lambda_n(G)$. Motivated by the Alon--Boppana bound and the Ramanujan phenomenon for regular graphs, Spielman conjectured that, for every graph $G$ with fixed average degree $d\ge 1$, its {\it Laplacian eigenratio} satisfies $$\frac{\lambda_2(G)}{\lambda_n(G)} \le \frac{d-2\sqrt{d-1}}{d+2\sqrt{d-1}}+o_n(1),$$ where $o_n(1)\to 0$ as $n\to\infty$. The main purpose of this paper is to investigate this conjecture. We show that the situation is mixed. On the negative side, the conjecture fails for infinitely many average degrees $d>2$, via constructions based on bipartite Ramanujan graphs. On the positive side, it holds in two important settings: we verify it for all average degrees $d\le 2$, and we prove it for all regular graphs. In fact, for regular graphs we obtain stronger bounds comparing higher Laplacian eigenvalues. As a consequence, we show that for every fixed $d\ge 3$ and every $\varepsilon>0$, every sufficiently large $d$-regular Ramanujan graph has linearly many adjacency eigenvalues below $-2\sqrt{d-1}+\varepsilon$, thereby strengthening earlier results of Li and Cioab\u{a} by giving an unconditional result of this form. We also settle two related conjectures: one of You and Liu concerning the maximum Laplacian eigenratio of trees, and one of Gu concerning the Hamiltonicity of graphs with large Laplacian eigenratio.

Joint with Quanyu Tang, Yuchang Wang and Zhiheng Zheng.