Existence and Explicit Constructions of q + 1 Regular Ramanujan Graphs for Every Prime Power q

For any prime power q, we give explicit constructions for many infinite linear families of q + 1 regular Ramanujan graphs. This partially solves a problem that was raised by A. Lubotzky, R. Phillips, and P. Sarnak. They gave the same results as here, but only for q being prime and not equal to two, and raised the question of the existence and explicit construction of such graphs for other degrees of regularity. Moreover, our construction removes the nondeterministic part of finding large prime numbers, which for some applications may appear in their construction. Our graphs are given as Cayley graphs of PGL2 or PSL2 over finite fields, with respect to very simple generators. They also satisfy all other extremal combinatorial properties that those of Lubotsky, Phillips, and Sarnak do.