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Ramanujan graphs and exponential sums over function fields

Sardari, Naser T. ; Zargar, Masoud


We prove that q + 1-regular Morgenstern Ramanujan graphs X-q,X-g (depending on g is an element of F-q[t]) have diameter at most (4/3 + epsilon) log(q) vertical bar X-q(,g)vertical bar + O-epsilon(1) (at least for odd q and irreducible g) provided that a twisted Linnik-Selberg conjecture over F-q(t) is true. This would break the 30 year-old upper bound of 2 log(q) vertical bar X-q(,g)vertical bar ...


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