Harmonic functions u : R → R are equivalent to integral manifolds of an exterior differential system with independence condition (M, I, ω). To this system one associates the space of conservation laws C. They provide necessary conditions for g : Sn−1 →M to be the boundary of an integral submanifold. We show that in a local sense these conditions are also sufficient to guarantee the existence of...

We study a class of holomorphic matrix models. The integrals are taken over middle dimensional cycles in the space of complex square matrices. As the size of the matrices tends to infinity, the distribution of eigenvalues is given by a measure with support on a collection of arcs in the complex planes. We show that the arcs are level sets of the imaginary part of a hyperelliptic integral connec...

We give a new geometrical interpretation of the local analytic solutions to a differential system, which we call a tautological system τ , arising from the universal family of Calabi-Yau hypersurfaces Ya in a G-variety X of dimension n. First, we construct a natural topological correspondence between relative cycles in Hn(X − Ya,∪D − Ya) bounded by the union of G-invariant divisors ∪D in X to t...

Suppose that G is an undirected graph whose edges have nonnegative integral capacities, that st, . . . ,s~ are k distinguished vertices in G, and that dti, I si<jl k, are nonnegative integral demands. We consider the problem of existence of an admissible multicommodity flow P= {Fii: 1 si<j= k) in which each Fii is a flow of value dii between si and sj. It is known that if kr5, then satisfying t...

Let C be the clutter of odd circuits of a signed graph ðG;SÞ: For nonnegative integral edge-weights w; we are interested in the linear program minðwtx: xðCÞ51; for C 2 C; and x50Þ; which we denote by (P). The problem of solving the related integer program clearly contains the maximum cut problem, which is NP-hard. Guenin proved that (P) has an optimal solution that is integral so long as ðG;SÞ ...

We study the integral uniform (multicommodity) flow problem in a graph G and construct a fractional solution whose properties are invariant under the action of a group of automorphisms Γ < Aut(G). The fractional solution is shown to be close to an integral solution (depending on properties of Γ), and in particular becomes an integral solution for a class of graphs containing Cayley graphs. As a...

We study the integral uniform (multicommodity) flow problem in a graph G and construct a fractional solution whose properties are invariant under the action of a group of automorphisms Γ < Aut(G). The fractional solution is shown to be close to an integral solution (depending on properties of Γ), and in particular becomes an integral solution for a class of graphs containing Cayley graphs. As a...

The Kontsevich integral of a knot is a graph-valued invariant which (when graded by the Vassiliev degree of graphs) is characterized by a universal property; namely it is a universal Vassiliev invariant of knots. We introduce a second grading of the Kontsevich integral, the Euler degree, and a geometric nullmove on the set of knots. We explain the relation of the null-move to S-equivalence, and...

A graph is called integral if all its eigenvalues (of the adjacency matrix) are integers. In this paper, the graphs K1,r •Kn , r ∗Kn , K1,r • Km,n , r ∗ Km,n and the tree K1,s • T (q, r,m, t) are defined. We determine the characteristic polynomials of these graphs and also obtain sufficient and necessary conditions for these graphs to be integral. Some sufficient conditions are found by using t...

2.1. Graphs and weights. A Kontsevich graph [2] is a directed graph with two types of vertices, type I and type II. The type I vertices are usually denoted 1, . . . ,m (m ≥ 0) and the type II vertices 1̄, . . . , n̄ (n ≥ 0). The directed edges are required to start at type I vertices. In fact, we will only consider the case m = 7, n = 2. A Lie graph is a Kontsevich graph with exactly two type II ...