Number-theoretical study on graph Laplacians

2014 Fiscal Year Final Research Report

• Project/Area Number: 23540014
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Nagoya Institute of Technology
• Principal Investigator: YAMAGISHI Masakazu 名古屋工業大学, 工学(系)研究科(研究院), 准教授 (40270996)
• Co-Investigator(Kenkyū-buntansha): MIZUSAWA Yasushi 名古屋工業大学, 大学院工学研究科, 准教授 (60453817)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Keywords: グラフのラプラシアン / チェビシェフ多項式 / 円分多項式 / 楕円曲線 / 有限体 / ライツアウトパズル / セルオートマトン / 双子素数

Outline of Final Research Achievements

We investigated number-theoretical behavior of the Laplacian for certain series of graphs, mainly for Cartesian products of path or cycle graphs. Specifically, we solved a problem of Hunziker-Machiavelo-Park in the affirmative way, conjectured a formula for the dimension of the space of harmonic functions in terms of Chebyshev polynomials, and succeeded in proving the conjecture.
Applying some techniques for polynomials used in the main study, we also obtained some results on a relation between Chebyshev polynomials and twin primes, on the resultant of Chebyshev polynomials and its application, and on isometric embeddings of finite fields.

Free Research Field

数論