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.
Dedicated to the memory of Jiří Matoušek
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement no. 227701.
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement no. 339109.
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Acknowledgements
We thank Bart Sevenster for helpful discussion on κ(G), and two referees for useful suggestions improving the presentation.
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Lovász, L., Schrijver, A. (2017). Nullspace Embeddings for Outerplanar Graphs. In: Loebl, M., Nešetřil, J., Thomas, R. (eds) A Journey Through Discrete Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-44479-6_23
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DOI: https://doi.org/10.1007/978-3-319-44479-6_23
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