论文标题
离散加权图的光谱预订和扰动
Spectral preorder and perturbations of discrete weighted graphs
论文作者
论文摘要
在本文中，我们介绍了具有磁性电势的加权图类别的几何和光谱预订关系。第一个预订是通过尊重磁力并实现重量某些不平等的图形同态的存在来表达的。第二个预订是指磁性加权图的相关拉板曲的光谱。这些关系对图表的基本和复合扰动的影响（删除边缘，收缩顶点等）对相应的laplacians的光谱进行了定量控制，从而将特征值的交织概括为概括。 我们给出了预订的几个应用：我们展示了如何根据这些预订进行分类，并证明了具有最大D-Clique的图表中某些特征值的稳定性。此外，我们在传递到跨度子图和相对于几何预订方面的磁性花颊常数的单调性时显示了特征值的单调性。最后，我们证明了一个精制的程序，可以检测无限覆盖图光谱中的光谱间隙。
In this article, we introduce a geometric and a spectral preorder relation on the class of weighted graphs with a magnetic potential. The first preorder is expressed through the existence of a graph homomorphism respecting the magnetic potential and fulfilling certain inequalities for the weights. The second preorder refers to the spectrum of the associated Laplacian of the magnetic weighted graph. These relations give a quantitative control of the effect of elementary and composite perturbations of the graph (deleting edges, contracting vertices, etc.) on the spectrum of the corresponding Laplacians, generalising interlacing of eigenvalues. We give several applications of the preorders: we show how to classify graphs according to these preorders and we prove the stability of certain eigenvalues in graphs with a maximal d-clique. Moreover, we show the monotonicity of the eigenvalues when passing to spanning subgraphs and the monotonicity of magnetic Cheeger constants with respect to the geometric preorder. Finally, we prove a refined procedure to detect spectral gaps in the spectrum of an infinite covering graph.
