Embedded in the Shadow of the Separator

Abstract

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.

The rotational dimension of a graph

“…Interest in fastest mixing Markov chains and graph conductivity led Boyd, Diaconis and Xiao [11] to investigate the same object (up to a trivial scaling). There and in [8] it was observed for connected G, that via semidefinite dualityâ(G) may also be expressed as the embedding problem |E| a(G) = maximize i∈N v i 2 subject to i∈N v i = 0, v i − v j ≤ 1 for ij ∈ E, v i ∈ R n for i ∈ N.…”

“…It was proven in [8] that for connected graphs G there always exist optimal solutions of (2) having dimension at most tree-width of G plus one (see Section 4 for the definition of tree-width). Intuitively, it is clear that the complexity of the structure of the optimal embedding mainly depends on some kind of central separator enclosing the barycenter in the optimal embedding.…”

“…The main results of [8] for (2) can be transferred, without too much difficulty, to the more general problem (3). To begin with, there is also a dual problem resembling (1).…”

“…Regarding the properties of optimal embeddings of (3) we will need a slightly sharpened version of the separator-shadow theorem of [8]. It describes a characteristic structural property of optimal solutions of (3) irrespective of dimensional considerations.…”

“…A second important result of [8] is the tree-width bound on the minimal dimension of optimal embeddings of (2). Its generalization to (3) yields an immediate bound on the rotational dimension of graphs.…”