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Explicit near-Ramanujan graphs of every degree

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arxiv 1909.06988 v3 pith:SQJ5ROZG submitted 2019-09-16 cs.DS cs.DMmath.CO

classification cs.DScs.DMmath.CO
keywords epsiloneverynear-ramanujanalgorithmboundedconstantdegreedeterministic
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abstract

For every constant $d \geq 3$ and $\epsilon > 0$, we give a deterministic $\mathrm{poly}(n)$-time algorithm that outputs a $d$-regular graph on $\Theta(n)$ vertices that is $\epsilon$-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by $2\sqrt{d-1} + \epsilon$ (excluding the single trivial eigenvalue of~$d$).

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  1. Parity families and a kernel-averaged L-function for near-Ramanujan signings

    math.CO 2026-07 reject novelty 6.0 of 10

    A parity-family averaging identity reduces the signed-spectral-radius problem to walk counting and yields claimed ε-versions of Bilu-Linial for dilute graphs, but the main theorems' final constant extraction is arithm...

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