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.

中文翻译：

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离散加权图的谱前序和扰动

在本文中，我们介绍了具有磁势的加权图类的几何和谱预序关系。第一个预序通过关于磁势和满足权重的某些不等式的图同态的存在来表示。第二个预序是指磁加权图的相关拉普拉斯算子的频谱。这些关系对图的基本和复合扰动（删除边、收缩顶点等）对相应拉普拉斯算子的谱的影响进行定量控制，概括了特征值的交错。我们给出了预序的几种应用：我们展示了如何根据这些预序对图进行分类，并且我们证明了具有最大 d-clique 的图中某些特征值的稳定性。而且，我们展示了传递到跨越子图时特征值的单调性以及磁性 Cheeger 常数相对于几何预序的单调性。最后，我们证明了一种改进的程序来检测无限覆盖图谱中的谱间隙。