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It was famously conjectured by Alon and proved by Friedman that if $d$ is fixed independent of $n$, then $\\lambda=2\\sqrt{d-1} +o(1)$ with high probability. In the present work we show that $\\lambda=O(\\sqrt{d})$ continues to hold with high probability as long as $d=O(n^{2/3})$, making progress towards a conjecture of Vu that the bound holds for all $1\\le d\\le n/2$. 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