Spielman's Laplacian eigenratio conjecture is disproved for infinitely many d>2 using Ramanujan graphs but verified for d≤2 and regular graphs, with stronger results and resolutions of two other conjectures.
On the second eigenvalue and linear expansion of regular graphs
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On Spielman's Laplacian Eigenratio Conjecture and Related Problems
Spielman's Laplacian eigenratio conjecture is disproved for infinitely many d>2 using Ramanujan graphs but verified for d≤2 and regular graphs, with stronger results and resolutions of two other conjectures.