The Laplacian Spectra of Graphs and Complex Networks

The study of graph Laplacian spectrum realized increasingly connections with many other areas. The objects arise in very diverse application, from combinatorial optimization to differential geometry, mathematical biology, computer science, machine learning, etc.) The Laplacian spectrum can be used to extract useful and important information about some graph invariants (for examples, expansion and vertex partition) that are hard to computer or estimate (see [1]). One of the most fascinating aspects of applications of eigenvalue methods in Combinatorics is the spectrum appear as a tool to prove results that appear to have nothing to do with the spectrum itself (see the excellent monographs [22]). The smallest nonzero and largest eigenvalues [39] can be expressed as solutions to a quadratic optimization problem. An important use of eigenvalues is Lovász's notation of the ϑ function ( [30]) which was initiated by problems in communication networks.

The graph Laplacian can be used in a number of ways to provide interesting geometric representations of a graph. One of the most work ( [14,22]) is the Colin de Verdière number of a graph, which is regarded as one of the most important recent developments in graph theory. The parameter, which is minor-monotone, determines the embeddability properties of graphs.

Alon and Milman [1] proved that the Laplacian spectrum of graphs plays a crucial role from the explicit constructions of expander graphs and superconcentrators to the design of various randomized algorithms. Based on the Laplacian eigenvalues and isoperimetric properties of graphs, Lubotzky, Phillips and Sarnak in [31] gave several family explicit constructions of good expander graphs.

The Laplacian matrix of graphs can be regarded as the discrete Laplacian operator on the differential manifolds. There is an interesting bilateral link between spectral graph theory and spectral Riemannian geometry. The concepts and methods of spectral geometry make it possible to obtain new results in the study of graph Laplacian spectrum. Conversely, the results in spectral graph theory may be transferred to the Laplacian operator on manifolds. We refer to the excellent book [10].

At the dawn of the new century, the power law networks (the scale-free networks) [3] and the small world networks [49] were discovered and studied. Since then, the analysis and modeling of networks, and networked dynamical system, have been the subject of considerable interdisciplinary interests, including physics, mathematics, computer science, biology, economics. The results have been called the “new science of networks” (see [48]).

The Laplacian spectra of a various way of modeling of networks which represent the real networks can be used to give a tentative classification scheme for empirical networks and provide useful insight into the relevant structural properties of real networks. Spectral techniques and methods based on the analysis of the largest eigenvalues and eigenvectors of some complex networks (for example the web networks [43]) have proven algorithmically successful in detecting communities and clusters. Mihail and Papadimitriou [37] showed that the largest eigenvalue of a power law graph with exponent β has power law distribution if the exponent β of the power law graph satisfies β > 3. Chung etc. in [12,13] established relationships between the spectrum of graphs and the structure of complex networks. It is showed that spectral methods are central in detecting clusters and finding patterns in various applications. For more information, we refer to the monograph [11] and a survey [43].

In this paper, we survey recent progress of the Laplacian spectra of graphs and complex networks. For basic information and earlier results, we refer several books and survey papers such as [10,11,33,34,38,39,52], for a detailed introduction and applications. The rest of this paper is organized as follows. In Section 2, some notations and results are presented. In Section 3, the Laplacian spectral radius of graphs with given degree sequences is extensively investigated. In Section 4, the relationship between the Laplacian coefficient and the ordering of graphs are studied, in particular, the Mohar’s problems [41] on the topic are discussed. In Section 5, we deeply studied the algebraic connectivity and structure and properties of the graph doubly stochastic matrix, in particular, on Merris’s problems [36] In Section 5, the Laplacian spectrum of random graphs and small world networks are discussed.

Let G = (V (G), E(G)) be a simple graph (no loops or multiple edges) with vertex set

, is the number of the edges incident with v i . Let D(G) = diag(d(u), u ∈ V ) be the diagonal matrix of vertex degrees of G and A(G) = (a ij ) be the (0, 1) adjacency matrix of G, where a ij = 1 for v i adjacent to v j and 0 for elsewhere. Then the matrix L(G) = D(G) -A(G) is called the Laplacian matrix of a graph G. It is obvious that L(G) is positive semidefinite

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