159 research outputs found
Spectral preorder and perturbations of discrete weighted graphs
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.Comment: 26 pages; 8 figure
Comparing Two Thickened Cycles: A Generalization of Spectral Inequalities
Motivated by an effort to simplify the Watts-Strogatz model for small-world networks, we generalize a theorem concerning interlacing inequalities for the eigenvalues of the normalized Laplacians of two graphs differing by a single edge. Our generalization allows weighted edges and certain instances of self loops. These inequalities were first proved by Chen et. al in [2] but our argument generalizes the simplified argument given by Li in [8]
Spectral distances on graphs
By assigning a probability measure via the spectrum of the normalized Laplacian to each graph and using Lp Wasserstein distances between probability measures, we define the corresponding spectral distances dp on the set of all graphs. This approach can even be extended to measuring the distances between infinite graphs. We prove that the diameter of the set of graphs, as a pseudo-metric space equipped with d1, is one. We further study the behavior of d1 when the size of graphs tends to infinity by interlacing inequalities aiming at exploring large real networks. A monotonic relation between d1 and the evolutionary distance of biological networks is observed in simulations
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