Visibility of quantum graph spectrum from the vertices

Essential spectrum of the discrete Laplacian on a perturbed periodic graph

We address the Laplacian on a perturbed periodic graph which might not be a periodic graph. We present a class of perturbed graphs for which the essential spectra of the Laplacians are stable even when the graphs are perturbed by adding and removing infinitely many vertices and edges. Using this result, we demonstrate how to determine the spectra of cone-like graphs, the upper-half plane, and graphs obtained from $\mathbb{Z}^2$ by randomly adding vertices.

具有指定谱的周期离散图

Periodic discrete graphs with prescribed spectrum

通过构造刷状几何的周期加权图，其离散拉普拉斯算子谱恰好有n个带隙，并证明通过适当选择权重可使带隙端点和谱上界达到指定值。

Minimum vertex covers and the spectrum of the normalized Laplacian on trees

We show that, in the graph spectrum of the normalized graph Laplacian on trees, the eigenvalue 1 and eigenvalues near 1 are strongly related to minimum vertex covers. In particular, for the eigenvalue 1, its multiplicity is related to the size of a minimum vertex cover, and zero entries of its eigenvectors correspond to vertices in minimum vertex covers; while for eigenvalues near 1, their distance to 1 can be estimated from minimum vertex covers; and for the largest eigenvalue smaller than 1, the sign graphs of its eigenvectors take vertices in a minimum vertex cover as representatives.

A Structured Inverse Spectrum Problem for Infinite Graphs and Unbounded Operators

Given an infinite graph $G$ on countably many vertices, and a closed, infinite set $Λ$ of real numbers, we prove the existence of an unbounded self-adjoint operator whose graph is $G$ and whose spectrum is $Λ$.