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254
The Laplacian spectrum of graphs
 Graph Theory, Combinatorics, and Applications
, 1991
"... Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, m ..."
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Cited by 151 (1 self)
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Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidthtype parameters of a graph. Some new results and generalizations are added. † This article appeared in “Graph Theory, Combinatorics, and Applications”, Vol. 2,
Some Applications of Laplace Eigenvalues of Graphs
 GRAPH SYMMETRY: ALGEBRAIC METHODS AND APPLICATIONS, VOLUME 497 OF NATO ASI SERIES C
, 1997
"... In the last decade important relations between Laplace eigenvalues and eigenvectors of graphs and several other graph parameters were discovered. In these notes we present some of these results and discuss their consequences. Attention is given to the partition and the isoperimetric properties of ..."
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Cited by 93 (0 self)
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In the last decade important relations between Laplace eigenvalues and eigenvectors of graphs and several other graph parameters were discovered. In these notes we present some of these results and discuss their consequences. Attention is given to the partition and the isoperimetric properties of graphs, the maxcut problem and its relation to semidefinite programming, rapid mixing of Markov chains, and to extensions of the results to infinite graphs.
Landscapes and Their Correlation Functions
, 1996
"... Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an additive const ..."
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Cited by 89 (15 self)
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Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an additive constant) eigenfuctions of a graph Laplacian. It is shown that elementary landscapes are characterized by their correlation functions. The correlation functions are in turn uniquely determined by the geometry of the underlying configuration space and the nearest neighbor correlation of the elementary landscape. Two types of correlation functions are investigated here: the correlation of a time series sampled along a random walk on the landscape and the correlation function with respect to a partition of the set of all vertex pairs.
Anisotropic Geometric Diffusion in Surface Processing
, 2000
"... INTRODUCTION Geometric evolution problems for curves and surfaces and especially curvature flow problems are an exciting and already classical mathematical research field. They lead to interesting systems of nonlinear partial differential equations and allow the appropriate mathematical modelling o ..."
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Cited by 87 (0 self)
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INTRODUCTION Geometric evolution problems for curves and surfaces and especially curvature flow problems are an exciting and already classical mathematical research field. They lead to interesting systems of nonlinear partial differential equations and allow the appropriate mathematical modelling of physical processes such as material interface propagation, fluid free boundary motion, crystal growth. On the other hand, curves and surfaces are essential objects in computer aided geometric design and computer graphics. Here, issues are fairing, modelling, deformation, and motion. Recently, geometric evolution problems and variational approaches have entered this research field as well and have turned out to be powerful tools. The aim of our work in the field of surface fairing and surface modelling is to modify "classical" curvature motion in a suitable way and apply it in computer graphics. 2 A GENERAL SCHEME Consider an image I :# R. A well known approach to image processing c
Analysis and Geometry on Configuration Spaces
, 1997
"... In this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds, namely, configuration spaces. More precisely, a differential geometry is introduced on the configuration space \Gamma X over a Riemannian manifold X ..."
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Cited by 50 (8 self)
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In this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds, namely, configuration spaces. More precisely, a differential geometry is introduced on the configuration space \Gamma X over a Riemannian manifold X. This geometry is "nonflat" even if X = IR d . It is obtained as a natural lifting of the Riemannian structure on X. In particular, a corresponding gradient r \Gamma , divergence div \Gamma , and LaplaceBeltrami operator H \Gamma = \Gammadiv \Gamma r \Gamma are constructed. The associated volume elements, i.e., all measures ¯ on \Gamma X w.r.t. which r \Gamma and div \Gamma become dual operators on L 2 (\Gamma X ; ¯), are identified as exactly the mixed Poisson measures with mean measure equal to a multiple of the volume element dx on X. In particular, all these measures obey an integration by parts formula w.r.t. vector fields on \Gamma X . The corresponding Dirichlet...
GraphDriven Features Extraction From Microarray Data
, 2002
"... Gene function prediction from microarray data is a first step toward better understanding the machinery of the cell from relatively cheap and easytoproduce data. In this paper we investigate whether the knowledge of many metabolic pathways and their catalyzing enzymes accumulated over the years ca ..."
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Cited by 43 (3 self)
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Gene function prediction from microarray data is a first step toward better understanding the machinery of the cell from relatively cheap and easytoproduce data. In this paper we investigate whether the knowledge of many metabolic pathways and their catalyzing enzymes accumulated over the years can help improve the performance of classifiers for this problem.
Computing Betti numbers via combinatorial Laplacians
 In Proc. 28th Ann. ACM Sympos. Theory Comput
, 1996
"... We use the Laplacian and power method to compute Betti numbers of simplicial complexes. This has a number of advantages over other methods, both in theory and in practice. It requires small storage space in many cases. It seems to run quickly in practice, but its running time depends on a ratio, ν, ..."
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Cited by 39 (0 self)
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We use the Laplacian and power method to compute Betti numbers of simplicial complexes. This has a number of advantages over other methods, both in theory and in practice. It requires small storage space in many cases. It seems to run quickly in practice, but its running time depends on a ratio, ν, of eigenvalues which we have yet to fully understand. We numerically verify a conjecture of Björner, Lovász, Vrećica, and ˘ Zivaljević on the chessboard complexes C(4,6), C(5,7), and C(5,8). Our verification suffers a technical weakness, which can be overcome in various ways; we do so for C(4,6) and C(5,8), giving a completely rigourous (computer) proof of the conjecture in these two cases. This brings up an interesting question in recovering an integral basis from a real basis of vectors. 1
Convergence of spectra of graphlike thin manifolds
 J. Geom. Phys
"... Abstract. We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the LaplaceBeltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at th ..."
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Cited by 37 (14 self)
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Abstract. We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the LaplaceBeltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand, if the the shrinking at the vertex parts of the manifold is sufficiently slower comparing to that of the edge parts, the limiting spectrum corresponds to decoupled edges with Dirichlet boundary conditions at the endpoints. At the borderline between the two regimes we have a third possibility when the limiting spectrum can be described by a nontrivial coupling at the vertices. 1.
SubGaussian estimates of heat kernels on infinite graphs
 Duke Math. J
, 2000
"... We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay. ..."
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Cited by 32 (10 self)
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We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.