Ramanujan graphs and expander families constructed from p-ary bent functions

We present a method for constructing an infinite family of non-bipartite Ramanujan graphs. We mainly employ p-ary bent functions of (p- 1) -form for this construction, where p is a prime number. Our result leads to construction of infinite families of expander graphs; this is due to the fact that Ramanujan graphs play as base expanders for constructing further expanders. For our construction we directly compute the eigenvalues of the Ramanujan graphs arsing from p-ary bent functions. Furthermore, we establish a criterion on the regularity of p-ary bent functions in m variables of (p- 1) -form when m is even. Finally, using weakly regular p-ary bent functions of ℓ-form, we find (amorphic) association schemes in a constructive way; this resolves the open case that ℓ= p- 1 for p> 2 for finding (amorphic) association schemes.