Ihara Zeta function, coefficients of Maclaurin series and Ramanujan graphs

Let X denote a connected (q + 1)-regular undirected graph of finite order n. The graph X is called Ramanujan whenever |λ|≤ 2q1 2 for all nontrivial eigenvalues λ of X. We consider the variant (u) of the Ihara Zeta function Z(u) of X defined by (u)-1= (1-u)(1-qu)(1-q1 2u)2n-2(1-u2)n(q-1) 2Z(u)if X is nonbipartite,(1-q2u2)(1-q1 2u)2n-4(1-u2)n(q-1) 2 +1Z(u) if X is bipartite. The function (u) satisfies the functional equation (q-1u-1) = (u). Let {hk}k=1∞ denote the number sequence given by d duln (q-1 2u) =k=0∞h k+1uk. In this paper, we establish the equivalence of the following statements: (i) X is Ramanujan; (ii) hk ≥ 0 for all k ≥ 1; (iii) hk ≥ 0 for infinitely many even k ≥ 2. Furthermore, we derive the Hasse-Weil bound for the Ramanujan graphs.