Noncommutative zeta functions of graphs and their applications

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K05249
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Hosei University (2017-2018); Utsunomiya University (2016)
• Principal Investigator: MITSUHASHI Hideo 法政大学, 理工学部, 教授 (60455095)
• Research Collaborator: SATO iwao
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: グラフのゼータ関数 / 四元数 / 量子ウォーク

Outline of Final Research Achievements

We defined several classes of weighted zeta functions of noncommutative weighted graphs; they are considered to have symmetric directed edges that are weighted by noncommutative quantities such as matrices or quaternions. We obtained main properties of the zeta functions such as determinant expressions. We generalized the theories of first and second weighted zeta functions of graphs to the case of quaternion-weighted graphs and applied them to the analysis of the spectra for quaternionic quantum walks on graphs. We also generalized the theory of first weighted zeta functions to much more general situation that includes the case of quaternions.

Free Research Field

群の表現論

Academic Significance and Societal Importance of the Research Achievements

本研究成果は，伊原ゼータ関数に始まるグラフのゼータ関数論を大きく発展させるものであり，四元数量子ウォークへの応用にもみられるように，離散数学，代数学，量子モデルなどへの多大な貢献を期待できる。四元数を初めとする非可換量がグラフ重みとして与えられた場合のゼータ関数のあり方を示したもので，グラフのゼータ関数の非可換化という前人未到のテーマへの礎となるものである。グラフの重み付きゼータ関数がグラフ上の量子ウォークの固有値問題に有効であることから，本研究も量子ウォークとの関連，さらには量子情報などとの関連が期待される。