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Abstract

We prove that $q+1$-regular Morgenstern Ramanujan graphs $X^{q,g}$ (depending
on $g\in\mathbb{F}_q[t]$) have diameter at most
$\left(\frac{4}{3}+\varepsilon\right)\log_{q}|X^{q,g}|+O_{\varepsilon}(1)$ (at
least for odd $q$ and irreducible $g$) provided that a twisted Linnik-Selberg
conjecture over $\mathbb{F}_q(t)$ is true. This would break the 30 year-old
upper bound of $2\log_{q}|X^{q,g}|+O(1)$, a consequence of a well-known upper
bound on the diameter of regular Ramanujan graphs proved by Lubotzky, Phillips,
and Sarnak using the Ramanujan bound on Fourier coefficients of modular forms.
We also unconditionally construct infinite families of Ramanujan graphs that
prove that $\frac{4}{3}$ cannot be improved.