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.

Imaginary replica analysis of loopy regular random graphs

Abstract: We present an analytical approach for describing spectrally constrained maximum entropy ensembles of finitely connected regular loopy graphs, valid in the regime of weak loop-loop interactions. We derive an expression for the leading two orders of the expected eigenvalue spectrum, through the use of infinitely many replica indices taking imaginary values. We apply the method to models in which the spectral constraint reduces to a soft constraint on the number of triangles, which exhibit `shattering' transitions to phases with extensively many disconnected cliques, to models with controlled numbers of triangles and squares, and to models where the spectral constraint reduces to a count of the number of adjacency matrix eigenvalues in a given interval. Our predictions are supported by MCMC simulations based on edge swaps with nontrivial acceptance probabilities.

Graph-theory treatment of one-dimensional strongly repulsive fermions

Abstract: The authors demonstrate that spectral graph theory can effectively describe strongly repulsive one-dimensional mixtures of ultracold fermions and circumvent the high computational complexity of this many-body system.

On the Largest Eigenvalue of Signless Laplacian Matrix of a Graph

Abstract: The signless Laplacian matrix of a graph is the sum of its diagonal matrix of vertex degrees and its adjacency matrix. Li and Feng gave some basic results on the largest eigenvalue and characteristic polynomial of adjacency matrix of a graph in 1979. In this paper, we translate these results into the signless Laplacian matrix of a graph and obtain the similar results. Denote the characteristic polynomials of A(G), K(G) and L(G) by �(G,�), �K(G,�) and �L(G,�), or simply by �(G), �K(G) andL(G), respectively. Since K(G) and L(G) are two real symmetric matrices, all of their eigenvalues are real. Write their largest eigenvalues byK(G) andL(G), respectively. For a long time, most scholars have been interested in the spectra of adjacency matrix and Laplacian matrix of a graph. Therefore, the two kinds of spectra are studied extensively in the literature. In (2), Cvetkovic, Doob and Sachs surveyed the properties and applications of

On eigenvectors of mixed graphs with exactly one nonsingular cycle

Abstract: Let G be a mixed graph. The eigenvalues and eigenvectors of G are respectively defined to be those of its Laplacian matrix. If G is a simple graph, [M. Fiedler: A property of eigenvectors of nonnegative symmetric matrices and its applications to graph theory, Czechoslovak Math. J. 25 (1975), 619–633] gave a remarkable result on the structure of the eigenvectors of G corresponding to its second smallest eigenvalue (also called the algebraic connectivity of G). For G being a general mixed graph with exactly one nonsingular cycle, using Fiedler’s result, we obtain a similar result on the structure of the eigenvectors of G corresponding to its smallest eigenvalue.