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 ” high-dimensional data that lies on a low-dimensional 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 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.

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 quantum-vs.-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 two-party 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.

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.

We prove that for each graph (G) (G) + 2, where and are minor-monotone 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 real-valued 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 ...

A graph G is intrinsically S 1-linked if for every embedding of the vertices of G into S 1, vertices that form the endpoints of two disjoint edges in G form a non-split link in the embedding. We show that a graph is intrinsically S 1 −linked if and only if it is not outer-planar. A graph is outer-flat if it can be embedded in the 3−ball such that all of its vertices map to the boundary of the 3−ball, all edges to the interior, and every cycle bounds a disk in the 3−ball that meets the graph only along its boundary. We show that a graph is outer-flat if and only if it is planar. 1