Göring, Frank ; Helmberg, Christoph ; Reiss, Susanna : Graph Realizations Associated with Minimizing the Maximum Eigenvalue of the Laplacian

Author(s):

Göring, Frank
Helmberg, Christoph
Reiss, Susanna

Title:

Graph Realizations Associated with Minimizing the Maximum Eigenvalue of the Laplacian

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 10, 2009

Mathematics Subject Classification:

05C50			[ Graphs and matrices ]
90C22			[ Semidefinite programming ]
90C35			[ Programming involving graphs or networks ]
05C10			[ Topological graph theory, imbedding ]
05C78			[ Graph labelling ]

Abstract:

In analogy to the absolute algebraic connectivity of Fiedler, we study the problem of minimizing the maximum eigenvalue of the Laplacian of a graph by redistributing the edge weights. Via semidefinite duality this leads to a graph realization problem in which nodes should be placed as close as possible to the origin while adjacent nodes must keep a distance of at least one. We prove three main results for a slightly generalized form of this embedding problem. First, given a set of vertices partitioning the graph into several or just one part, the barycenter of each part is embedded on the same side of the affine hull of the set as the origin. Second, there is an optimal realization of dimension at most the tree-width of the graph plus one and this bound is best possible in general. Finally, bipartite graphs possess a one dimensional optimal embedding.

Keywords:

spectral graph theory, semidefinite programming, eigenvalue optimization, embedding, graph partitioning, tree-width

Language:

English

Publication time:

5 / 2009