Zusammenfassung
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 ...
Zusammenfassung
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 + 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 4/3 scannot be improved. (C) 2020 Elsevier Inc. All rights reserved.
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