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.

Given a connected graph G = (N, E) with node weights s ∈ R N + and nonnegative edge lengths, we study the following embedding problem related to an eigenvalue optimization problem over the second smallest eigenvalue of the (scaled) Laplacian of G: Find vi ∈ R |N| , i ∈ N so that distances between adjacent nodes do not exceed prescribed edge lengths, the weighted barycenter of all points is at the origin, and P i∈N si vi 2 is maximized. In the case of a two dimensional optimal solution this corresponds to the equilibrium position of a quickly rotating net consisting of weighted mass points that are linked by massless cables of given lengths. We define the rotational dimension of G to be the minimal dimension k so that for all choices of lengths and weights an optimal solution can be found in R k and show that this is a minor monotone graph parameter. We give forbidden minor characterizations up to rotational dimension 2 and prove that the rotational dimension is always bounded above by the tree-width of G plus one.

A unique-maximum k-coloring with respect to faces of a plane graph G is a coloring with colors 1,. .. , k so that, for each face α of G, the maximum color occurs exactly once on the vertices of α. We prove that any plane graph is unique-maximum 3-colorable and has a proper unique-maximum coloring with 6 colors.

Menger's Theorem for digraphs states that for any two vertex sets A and B of a digraph D such that A cannot be separated from B by a set of at most t vertices, there are t +1 disjoint A±B-paths in D. Here a short and elementary proof of a more general theorem is given.

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