Ihara zeta function, coefficients of Maclaurin series, and Ramanujan graphs

Abstract

Let $X$ denote a connected $(q+1)$-regular undirected graph of finite order $n$. The graph $X$ is called Ramanujan whenever $$|\lambda |\le 2q^{\frac{1}{2}}$$ for all nontrivial eigenvalues $\lambda$ of $X$. We consider the variant $\Xi (u)$ of the Ihara zeta function $Z(u)$ of $X$ defined by \begin{gather*} Ξ(u)^{-1} = \left\{ \begin{array}{ll} (1-u)(1-qu)(1-q^{\frac{1}{2}} u)^{2n-2}(1-u^2)^{\frac{n(q-1)}{2}} Z(u) \qquad &\hbox{if $X$ is nonbipartite}, (1-q^2u^2) (1-q^{\frac{1}{2}} u)^{2n-4} (1-u^2)^{\frac{n(q-1)}{2}+1} Z(u) \qquad &\hbox{if $X$ is bipartite}. \end{array} \right. \end{gather*} The function $\Xi (u)$ satisfies the functional equation $\Xi (q^{-1}{u}^{-1})=\Xi (u)$. Let $\{h_k{\}}_{k=1}^{\infty }$ denote the number sequence given by $$\frac{d}{du}\ln \Xi (q^{-\frac{1}{2}}u)=\sum _{k=0}^{\infty }{h}_{k+1}{u}^k.$$ In this paper we establish the equivalence of the following statements: (i) $X$ is Ramanujan; (ii) $h_k\ge 0$ for all $k\ge 1$; (iii) $h_k\ge 0$ for infinitely many even $k\ge 2$. Furthermore we derive the Hasse--Weil bound for the Ramanujan graphs.

8 pages