The rotational dimension of a graph

Abstract

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.

On Minimizing the Spectral Width of Graph Laplacians and Associated Graph Realizations

“…An almost identical result holds for optimal U of E λn−λ2 if some special graphs are excluded (Theorem 4.7). The last two theorems allow us to transfer the structural results of [23,24,22] linking optimal graph realizations (and the associated eigenvectors) to the separator structure of the graph: optimal realizations V fold inward along separators (Corollary 4.3), optimal realizations U fold outward along separators (Corollary 4.8), and there exist optimal realizations where both realizations have dimension bounded by the tree-width of the graph plus one (Corollary 4.5, Theorem 4.9, and Corollary 4.10). Like in [29], for unweighted Laplacians the eigenvectors have an interpretation as a realization where one also optimizes over the edge length parameters l (Theorems 7.1 and 7.2).…”

“…Properties common to the coupled and the single problems. Graph realizations induced by optimal solutions of E λ2 and E λn are tightly linked to the separator structure of the graph; see [24] and [22]. The aim of this section is to investigate which of the properties of the single problems can be saved for the combined problem E λn−λ2 .…”

“…49.23.145. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php [23,24] and [22]). …”

“…Given a graph, we consider best distributions of nonnegative weights on its edges that minimize this width and study their relation to structural properties of the graph as well as to the associated eigenvalues and eigenvectors of the corresponding weighted Laplacian. The problem of minimizing λ n (L w )−λ 2 (L w ) combines the problems of maximizing λ 2 (L w ) studied in [17,23,24] and minimizing λ n (L w ) investigated in [16,22,29]. Our work builds heavily on the semidefinite optimization approaches in [23,24,22] which offer an intriguing interpretation of the semidefinite dual as a (nonstandard) graph embedding or graph realization problem with the property that optimal realizations give rise to geometric visualizations of eigenvectors corresponding to the optimized eigenvalue.…”

A metric for layout-friendly microarchitecture optimization in high-level synthesis

“…Spreading score is indirectly related to cut size as well, as suggested by the following result from [12]. For a simplified version of the problem with wi = 1 and lij = 1, the dual problem of the relaxation in Eqn.…”