A study on zeta functions of graphs via harmonic analysis and its applications

2021 Fiscal Year Final Research Report

• Project/Area Number: 18K03242
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Ehime University
• Principal Investigator: Yamasaki Yoshinori 愛媛大学, 理工学研究科(理学系), 教授 (00533035)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Keywords: Iharaゼータ関数 / Ramanujanグラフ / 四元数環

Outline of Final Research Achievements

For each finite graph, we have the associated Ihara zeta function from which we can know various properties of the underlying graph. In this study, we first tried to construct new family of Ramanujan graphs which are important from the viewpoint of applications and characterized by the fact that the associated Ihara zeta function satisfies an analogue of the Riemann hypothesis. Specifically, we constructed graphs via quaternion algebras and their orders, and showed that, in some special cases, they are actually Ramanujan. Moreover, we also studied the logarithmic derivative of Ihara zeta function with a representation of the fundamental group and, in some special cases, explicitly calculated the Taylor coefficients at the origin of that, which reflect various features of the underlying graph such as "complexity".

Free Research Field

解析数論

Academic Significance and Societal Importance of the Research Achievements

Ramanujanグラフは、ハッシュ関数の構成など現在暗号理論の分野においても応用が著しい。それゆえ本研究で構成したRamanujanグラフは実社会での活用に直結する可能性が期待できる。また、本グラフはパラメータ付きで構成されているため、必要に応じて適宜パラメータを調整して利用できる点も利点の一つであると考えられる。また、Iharaゼータ関数の対数微分の研究は、グラフ理論とゼータ関数論の境界領域に位置するあまり前例がない研究であるため、特別な場合ではあるがここで一般論を整理・展開できたことは、両者の境界領域のすそ野を広げるという意味でも非常に有意義であると考えられる。