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Nullspace Embeddings for Outerplanar Graphs

  • László Lovász
  • Alexander Schrijver
Chapter

Abstract

We study relations between geometric embeddings of graphs and the spectrum of associated matrices, focusing on outerplanar embeddings of graphs. For a simple connected graph G = (V, E), we define a “good” G-matrix as a V × V matrix with negative entries corresponding to adjacent nodes, zero entries corresponding to distinct nonadjacent nodes, and exactly one negative eigenvalue. We give an algorithmic proof of the fact that if G is a 2-connected graph, then either the nullspace representation defined by any “good” G-matrix with corank 2 is an outerplanar embedding of G, or else there exists a “good” G-matrix with corank 3.

Notes

Acknowledgements

We thank Bart Sevenster for helpful discussion on κ(G), and two referees for useful suggestions improving the presentation.

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Copyright information

© Springer International publishing AG 2017

Authors and Affiliations

  1. 1.Eötvös Loránd UniversityBudapestHungary
  2. 2.University of Amsterdam and CWIAmsterdamThe Netherlands

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