Ramanujan graphs and exponential sums over function fields

We prove that $q+1$-regular Morgenstern Ramanujan graphs $X^{q,g}$ (depending on $g\in {\mathbb{F}}_q[t]$) have diameter at most $(\frac{4}{3}+\epsilon ){\log }_q\mathrm{\mid }{X}^{q,g}\mathrm{\mid }+O_{\epsilon }(1)$ (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 $2{\log }_q\mathrm{\mid }{X}^{q,g}\mathrm{\mid }+O(1)$, 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.