Spectral graph theory

About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.

Network Cluster-Robust Inference

Abstract: Since network data commonly consists of observations on a single large network, researchers often partition the network into clusters in order to apply cluster-robust inference methods. All existing such methods require clusters to be asymptotically independent. We prove under mild conditions that, in order for this requirement to hold for network-dependent data, it is necessary and sufficient for clusters to have low conductance, the ratio of edge boundary size to volume. This yields a simple measure of cluster quality. We find in simulations that, when clusters have low conductance, cluster-robust methods outperform HAC estimators in terms of size control. However, for important classes of networks lacking low-conductance clusters, the methods can exhibit substantial size distortion. To assess the existence of low-conductance clusters and construct them, we draw on results in spectral graph theory that connect conductance to the spectrum of the graph Laplacian. Based on these results, we propose to use the spectrum to determine the number of low-conductance clusters and spectral clustering to construct them.

Some notes on the spectral perturbations of the signless Laplacian of a graph

Abstract: Let G be a simple graph and let Q (G) be the signless Laplacian matrix of G. In this paper we obtain some results on the spectral perturbation of the matrix Q (G) under an edge addition or an edge contraction.

Another form of the transmission function

Abstract: The transmission function describes the passage of the electric current from one point of an electric circuit to another. By now, this is also applied to molecules which are potential candidates for uses in the molecular electronics. We mean the modern branch of electronics which has a goal of reducing the sizes of its devices down to molecular ones and planning indeed to apply single molecules as conducting wires and functional components of microcircuits. For calculating the transmission function, some authors utilize the well-known idea of representing a molecule by a (molecular) graph, which allows them to apply for treating the latter also powerful methods of spectral graph theory. For instance, we refer to the paper by Fowler et al. (Chem Phys Lett. 465 2008) 142–146, where one such expression for this function is given. Our objective is to demonstrate that the same calculational result can be obtained using a different set of characteristic polynomials of graphs (which also slightly reduces a mathematical notation). Specifically, we apply one theorem of Kolmykov to the basic formula derived by these authors.

Spectral Properties of Adjacency and Distance Matrices for Various Networks

Abstract: The spectral properties of the adjacency (connectivity) and distance matrix for various types of networks: exponential, scale-free (Albert---Barabasi) and classical random ones (Erdős---Renyi) are evaluated. The graph spectra for dense graph in the Erdős---Renyi model are derived analytically.