A method to improve the lower bounds for Ramsey numbers R(k,l) is provided: one may construct cyclic graphs by using cubic residues modulo the primes in the form p = 6m + 1 to produce desired examples. In particular, we obtain 16 new lower bounds, which are R(6,12) ≥ 230, R(5,15) ≥ 242, R(6,14) ≥ 284, R(6,15) ≥ 374, R(6,16) ≥ 434, R(6,17) ≥ 548, R(6,18) ≥ 614, R(6,19) ≥ 710, R(6,20) ≥ 878, R(6,21) ≥ 884, R(7,19) ≥ 908, R(6,22) ≥ 1070, R(8,20) ≥ 1094, R(7,21) ≥ 1214, R(9,20) ≥ 1304, R(8,21) ≥ 1328.
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