Yusuke Higuchi

Spectral structure of the Laplacian on a covering graph

In this paper we study the spectral structure of the discrete Laplacian on an infinite graph. We show that, for a finite graph including a certain kind of a family of cycles, the spectrum of the Laplacian on its homology universal covering graph has band structure and no eigenvalues; furthermore it is purely absolutely continuous. Interesting examples that illustrate our theorems are also exhibited.

Boundary Area Growth and the Spectrum of Discrete Laplacian

We introduce the boundary area growth as a new quantity for an infinite graph. Using this, we give some upper bounds for the bottom of the spectrum of the discrete Laplacian which relates closely to the transition operator. We also give some applications and examples.

On the spectrum of a discrete Laplacian on ℤ with finitely supported potential

We study spectral properties of the discrete Laplacian L = −Δ + V on ℤ with finitely supported potential V. We give sufficient and necessary conditions for L to satisfy that the number of negative (resp. positive) eigenvalues is equal to one of the points x on which V(x) is negative (resp. positive). In addition, we prove that L has at least one discrete eigenvalue. If ∑ x∈ℤ V(x) = 0, then L has both negative and positive discrete eigenvalues.

Quantum walks induced by Dirichlet random walks on infinite trees

We consider the Grover walk on infinite trees from the view point of spectral analysis. From the previous works, infinite regular trees provide localization. In this paper, we give the complete characterization of the eigenspace of this Grover walk, which involves localization of its behavior and recovers the previous works. Our result suggests that the Grover walk on infinite trees may be regarded as a limit of the quantum walk induced by the isotropic random walk with the Dirichlet boundary condition at the

Non-separating 2-factors of an even-regular graph

n

Combinatorial curvature for planar graphs

-th depth rather than one with the Neumann boundary condition.

Spectral and asymptotic properties of Grover walks on crystal lattices

For a 2-factor F of a connected graph G, we consider G-F, which is the graph obtained from G by removing all the edges of F. If G-F is connected, F is said to be a non-separating 2-factor. In this paper we study a sufficient condition for a 2r-regular connected graph G to have such a 2-factor. As a result, we show that a 2r-regular connected graph G has a non-separating 2-factor whenever the number of vertices of G does not exceed 2r^2+r.