2019/05/31 by Hau-Wen Huang, Huang, Hau-Wen
Chemistry · Mathematics · #05C50 #11M26 #Advanced Algebra and Geometry #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Synthesis and Properties of Aromatic Compounds
paper · pdf · doi:10.48550/arxiv.1905.13485
openalex publication_date 2019/05/31 · openalex created_date 2020/07/10 · openalex updated_date 2026/07/28
Let X denote a connected (q+1)-regular undirected graph of finite order n. The graph X is called Ramanujan whenever |λ|≤ 2q(1)/(2) for all nontrivial eigenvalues λ of X. We consider the variant Ξ(u) of the Ihara zeta function Z(u) of X defined by Ξ(u)-1 = \ (1-u)(1-qu)(1-q(1)/(2) u)2n-2(1-u2)(n(q-1))/(2) Z(u) · amp;\hboxif X is nonbipartite, (1-q2u2) (1-q(1)/(2) u)2n-4 (1-u2)(n(q-1))/(2)+1 Z(u) · amp;\hboxif X is bipartite. . The function Ξ(u) satisfies the functional equation Ξ(q-1 u-1)=Ξ(u). Let \hk\k=1^∞ denote the number sequence given by (d)/(du)ln Ξ(q-(1)/(2)u) =∑k=0^∞ hk+1 uk. 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.