AbstractGeneralized Paley graphs are cyclic graphs constructed from quadratic or higherresidues of finite fields. Using this type of cyclic graphs to study the lower boundsfor classical Ramsey numbers, has high computing efficiency in both looking forparameter sets and computing clique numbers. We have found a new generaliza-tion of generalized Paley graphs, i.e. automorphism cyclic graphs, also having thesame advantages. In this paper we study the properties of the parameter sets ofautomorphism cyclic graphs, and develop an algorithm to compute the order of themaximum independent set, based on which we get new lower bounds for 8 classicalRamsey numbers: R(3, 22) > 131, R(3, 23) > 137, R(3, 25) > 154, R(3, 28) > 173,R(3, 29) > 184, R(3, 30) > 190, R(3, 31) > 199, R(3, 32) > 214. Furthermore, wealso get R(5, 23) > 521 based on R(3, 22) > 131. These nine results above improvetheir corresponding best known lower bounds...