On Characterisations and Classifications of graphs by their Eigenvalues

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K05263
• 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: Hiroshima Institute of Technology
• Principal Investigator: Taniguchi Tetsuji 広島工業大学, 工学部, 准教授 (90543728)
• Co-Investigator(Kenkyū-buntansha): 宗政 昭弘 東北大学, 情報科学研究科, 教授 (50219862)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 代数的グラフ理論 / スペクトラルグラフ理論 / 代数的組合せ論

Outline of Final Research Achievements

In this project, we have been studying on the structure of a generalized Bethe tree and a Hoffman graph. The structure of a generalized Bethe tree is hierarchical and symmetric, but it is very difficult to see the eigenvalues. With Segawa, Kubota, and Yoshie, we obtained results on the periodicity of the Grover Walk in the class.
In addition, by generalizing the fat vertices and (signed) edges of a Hoffman graph, the Hoffman graph and generalized integral lattices can be associated. This makes it possible to explain the ingerability of generalized integral lattices. In problem of classifications of 3-lattices, we obtained the results that can be said to be the final goal on the line graph of the signed graph derived from integral lattices.

Free Research Field

代数的グラフ理論

Academic Significance and Societal Importance of the Research Achievements

量子ウォークや整格子の分類・特徴付けは，量子コンピュータ，代数符号などの発展に貢献できる。
量子ウォークの周期性などは，量子探索などビッグデータの発展に寄与できる。また，グラフという構造は代数構造を詳細に表す言葉として重要である。ホフマングラフの一般化を得たことは，これまで整格子の構造の詳細を説明する道具がほとんどなかった中で，大きな成果と言ってよい。