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The condition for a connected $(q+1)$-regular graph to be Ramanujan is that every nonzero eigenvalue $\lambda$ of the graph Laplacian satisfy $$q+1-2\sqrt{q}\le \lambda\le q+1+2\sqrt{q}.$$ With a little bit of manipulation, we can rewrite this condition as the polynomial inequality $$\lambda^{2}+q^{2}-2\lambda q-2\lambda-2q+1\ \le0$$ which surprisingly is symmetric in $\lambda$ and $q$, despite the completely different role played by $q$ and $\lambda$ here.

This might just be a coincidence, but I think it's surprising enough to ponder whether there's anything more going on that would give a conceptual reason for this symmetry.

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    $\begingroup$ The inequality can also be rewritten as $|\sqrt{\lambda}-\sqrt{q}|\leq 1$. $\endgroup$
    – GH from MO
    Commented Aug 5, 2022 at 23:41
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    $\begingroup$ I hadn't seen this version of the definition of a Ramanujan graph before, but I note that the inequality you wrote down is formally the same as the inequality in Hasse's theorem on the number of points on an elliptic curve over a finite field with $q$ elements. In the elliptic curve case, there is a connection with the Riemann hypothesis for function fields, and in the Ramanujan graph case, there is a connection with the Riemann hypothesis for the Ihara zeta function. $\endgroup$
    – Timothy Chow
    Commented Aug 6, 2022 at 12:39
  • $\begingroup$ I am aware of the connection with Ihara/Hasse-Weil zeta function, but there is no apparent symmetry in the definition of these functions neither. $\endgroup$
    – Antoine Labelle
    Commented Aug 6, 2022 at 13:33
  • $\begingroup$ @AntoineLabelle Do you understand why the same inequality arises for Ramanujan graphs and for counting points on elliptic curves? If so, that might be worth spelling out as part of your question. $\endgroup$
    – Timothy Chow
    Commented Aug 7, 2022 at 9:07
  • $\begingroup$ To me this reads like the truth of the analogue of the Riemann Hypothesis for some L-function implying its truth for a whole related family of L-functions, which may be closed under some operation arising for the regular graph structure. $\endgroup$
    – Sylvain JULIEN
    Commented Aug 8, 2022 at 10:37

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