Arbitrary Spectral Edge of Regular Graphs

We prove that for each $d\geq 3$ and $k\geq 2$, the set of limit points of the first $k$ eigenvalues of sequences of $d$-regular graphs is \[ \{(μ_1,\dots,μ_k): d=μ_1\geq \dots\geq μ_{k}\geq2\sqrt{d-1}\}. \] The result for $k=2$ was obtained by Alon and Wei, and our result confirms a conjectur