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16
The fastest mixing Markov process on a graph and a connection to a maximum variance unfolding problem
 SIAM REVIEW
, 2006
"... We consider a Markov process on a connected graph, with edges labeled with transition rates between the adjacent vertices. The distribution of the Markov process converges to the uniform distribution at a rate determined by the second smallest eigenvalue λ2 of the Laplacian of the weighted graph. I ..."
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Cited by 44 (4 self)
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We consider a Markov process on a connected graph, with edges labeled with transition rates between the adjacent vertices. The distribution of the Markov process converges to the uniform distribution at a rate determined by the second smallest eigenvalue λ2 of the Laplacian of the weighted graph. In this paper we consider the problem of assigning transition rates to the edges so as to maximize λ2 subject to a linear constraint on the rates. This is the problem of finding the fastest mixing Markov process (FMMP) on the graph. We show that the FMMP problem is a convex optimization problem, which can in turn be expressed as a semidefinite program, and therefore effectively solved numerically. We formulate a dual of the FMMP problem and show that it has a natural geometric interpretation as a maximum variance unfolding (MVU) problem, i.e., the problem of choosing a set of points to be as far apart as possible, measured by their variance, while respecting local distance constraints. This MVU problem is closely related to a problem recently proposed by Weinberger and Saul as a method for “unfolding ” highdimensional data that lies on a lowdimensional manifold. The duality between the FMMP and MVU problems sheds light on both problems, and allows us to characterize and, in some cases, find optimal solutions.
The Minimum Rank of Symmetric Matrices Described by a Graph: A Survey
, 2007
"... The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i ̸ = j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. This paper surveys the current state of knowledge on the problem of determining the minim ..."
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Cited by 43 (18 self)
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The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i ̸ = j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. This paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues.
A variant on the graph parameters of Colin de Verdière: implications to the minimum rank of graphs
 J. LINEAR ALGEBRA
, 2005
"... For a given undirected graph G, the minimum rankof G is defined to be the smallest possible rankover all real symmetric matrices A whose (i, j)th entry is nonzero whenever i � = j and {i, j} is an edge in G. Building upon recent workinvolving maximal coranks (or nullities) of certain symmetric mat ..."
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Cited by 14 (9 self)
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For a given undirected graph G, the minimum rankof G is defined to be the smallest possible rankover all real symmetric matrices A whose (i, j)th entry is nonzero whenever i � = j and {i, j} is an edge in G. Building upon recent workinvolving maximal coranks (or nullities) of certain symmetric matrices associated with a graph, a new parameter ξ is introduced that is based on the corankof a different but related class of symmetric matrices. For this new parameter some properties analogous to the ones possessed by the existing parameters are verified. In addition, an attempt is made to apply these properties associated with ξ to learn more about the minimum rankof graphs – the original motivation.
Spectral graph theory and the inverse eigenvalue of a graph. Presentation at
 Directions in Combinatorial Matrix Theory Workshop, Banff International Research Station
, 2004
"... Abstract. Spectral Graph Theory is the study of the spectra of certain matrices defined from a given graph, including the adjacency matrix, the Laplacian matrix and other related matrices. Graph spectra have been studied extensively for more than fifty years. In the last fifteen years, interest has ..."
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Cited by 12 (2 self)
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Abstract. Spectral Graph Theory is the study of the spectra of certain matrices defined from a given graph, including the adjacency matrix, the Laplacian matrix and other related matrices. Graph spectra have been studied extensively for more than fifty years. In the last fifteen years, interest has developed in the study of generalized Laplacian matrices of a graph, that is, real symmetric matrices with negative offdiagonal entries in the positions described by the edges of the graph (and zero in every other offdiagonal position). The set of all real symmetric matrices having nonzero offdiagonal entries exactly where the graph G has edges is denoted by S(G). Given a graph G, the problem of characterizing the possible spectra of B, such that B ∈S(G), has been referred to as the Inverse Eigenvalue Problem of a Graph. In the last fifteen years a number of papers on this problem have appeared, primarily concerning trees. The adjacency matrix and Laplacian matrix of G and their normalized forms are all in S(G). Recent work on generalized Laplacians and Colin de Verdière matrices is bringing the two areas closer together. This paper surveys results in Spectral Graph Theory and the Inverse Eigenvalue Problem of a Graph, examines the connections between these problems, and presents some new results on construction of a matrix of minimum rank for a given graph having a special form such as a 0,1matrix or a generalized Laplacian.
Nondeterministic quantum query and communication complexities, to appear in
 version in Proc. IEEE Complexity '2000. cs.CC/0001014
"... Abstract. We study nondeterministic quantum algorithms for Boolean functions f. Such algorithms have positive acceptance probability on input x iff f(x) = 1. In the setting of query complexity, we show that the nondeterministic quantum complexity of a Boolean function is equal to its “nondeterminis ..."
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Cited by 11 (0 self)
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Abstract. We study nondeterministic quantum algorithms for Boolean functions f. Such algorithms have positive acceptance probability on input x iff f(x) = 1. In the setting of query complexity, we show that the nondeterministic quantum complexity of a Boolean function is equal to its “nondeterministic polynomial ” degree. We also prove a quantumvs.classical gap of 1 vs. n for nondeterministic query complexity for a total function. In the setting of communication complexity, we show that the nondeterministic quantum complexity of a twoparty function is equal to the logarithm of the rank of a nondeterministic version of the communication matrix. This implies that the quantum communication complexities of the equality and disjointness functions are n + 1 if we do not allow any error probability. We also exhibit a total function in which the nondeterministic quantum communication complexity is exponentially smaller than its classical counterpart.
Forbidden minors for the class of graphs G with ξ(G
"... Abstract. For a given simple graph G, S(G) is defined to be the set of real symmetric matrices A whose (i, j)th entry is nonzero whenever i = j and ij is an edge in G. In [2], ξ(G) is defined to be the maximum corank (i.e., nullity) among A ∈ S(G) having the Strong Arnold Property; ξ is used to stu ..."
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Cited by 10 (4 self)
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Abstract. For a given simple graph G, S(G) is defined to be the set of real symmetric matrices A whose (i, j)th entry is nonzero whenever i = j and ij is an edge in G. In [2], ξ(G) is defined to be the maximum corank (i.e., nullity) among A ∈ S(G) having the Strong Arnold Property; ξ is used to study the minimum rank/maximum eigenvalue multiplicity problem for G. Since ξ is minor monotone, the graphs G such that ξ(G) ≤ k can be described by a finite set of forbidden minors. We determine the forbidden minors for ξ(G) ≤ 2 and present an application of this characterization to computation of minimum rank among matrices in S(G).
Embedded in the Shadow of the Separator
, 2005
"... We study the problem of maximizing the second smallest eigenvalue of the Laplace matrix of a graph over all nonnegative edge weightings with bounded total weight. The optimal value is the absolute algebraic connectivity introduced by Fiedler, who proved tight connections of this value to the connect ..."
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Cited by 5 (1 self)
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We study the problem of maximizing the second smallest eigenvalue of the Laplace matrix of a graph over all nonnegative edge weightings with bounded total weight. The optimal value is the absolute algebraic connectivity introduced by Fiedler, who proved tight connections of this value to the connectivity of the graph. Using semidefinite programming techniques and exploiting optimality conditions we show that the problem is equivalent to finding an embedding of the n nodes in n−space so that their barycenter is at the origin, the distance between adjacent nodes is bounded by one and the nodes are spread as much as possible (the sum of the squared norms is maximized). For connected graphs we prove that for any separator in the graph, at least one of the two separated node sets is embedded in the shadow (with the origin being the light source) of the convex hull of the separator. In particular, the barycenters of partitions induced by separators are separated by the affine subspace spanned by the nodes of the separator. Furthermore, we show that there always exists an optimal embedding whose dimension is bounded by the tree width of the graph plus one.
On the Relation Between Two MinorMonotone Graph Parameters
, 1997
"... We prove that for each graph (G) (G) + 2, where and are minormonotone graph invariants introduced by Colin de Verdi'ere [3] and van der Holst, Laurent and Schrijver [5]. It is also shown that a graph G exists with (G) ! (G). The graphs G with maximal planar complement and (G) = jV (G)j \Gamm ..."
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Cited by 4 (0 self)
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We prove that for each graph (G) (G) + 2, where and are minormonotone graph invariants introduced by Colin de Verdi'ere [3] and van der Holst, Laurent and Schrijver [5]. It is also shown that a graph G exists with (G) ! (G). The graphs G with maximal planar complement and (G) = jV (G)j \Gamma 4, characterised by Kotlov, Lov'asz and Vempala, are shown to be forbidden minors for fH j (H) ! jV (G)j \Gamma 4g. Introduction Given a graph G = (V; E) without loops or multiple edges, define OG as the collection of realvalued symmetric V \Theta V matrices M = (m ij ) satisfying 1. if ij 2 E, then m ij ! 0, and 2. if ij 62 E and i 6= j, then m ij = 0. There is no restriction on the diagonal entries. The elements of OG are sometimes called discrete Schrodinger operators. A matrix M 2 OG satisfies the Strong Arnol'd Hypothesis, SAH for short, if there is no nonzero symmetric matrix X = (x ij ) such that MX = 0, and such that x ij = 0 whenever i = j or ij 2 E. By i (M) we denote ...
Spectral algorithms
, 2009
"... Summary. Spectral methods refer to the use of eigenvalues, eigenvectors, singular values and singular vectors. They are widely used in Engineering, Applied Mathematics and Statistics. More recently, spectral methods have found numerous applications in Computer Science to “discrete ” as well “continu ..."
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Cited by 4 (0 self)
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Summary. Spectral methods refer to the use of eigenvalues, eigenvectors, singular values and singular vectors. They are widely used in Engineering, Applied Mathematics and Statistics. More recently, spectral methods have found numerous applications in Computer Science to “discrete ” as well “continuous” problems. This book describes modern applications of spectral methods, and novel algorithms for estimating spectral parameters. In the first part of the book, we present applications of spectral methods to problems from a variety of topics including combinatorial optimization, learning and clustering. The second part of the book is motivated by efficiency considerations. A feature of many modern applications is the massive amount of input data. While sophisticated algorithms for matrix computations have been developed over a century, a more recent development is algorithms based on “sampling on the fly ” from massive matrices. Good estimates of singular values and low rank approximations of the whole matrix can be provably derived from a sample. Our