TY - JOUR

T1 - Discrete trace theorems and energy minimizing spring embeddings of planar graphs

AU - Urschel, John C.

AU - Zikatanov, Ludmil T.

N1 - Funding Information:
The work of L. Zikatanov was supported in part by NSF grants DMS-1720114 and DMS-1819157 . The work of J. Urschel was supported in part by ONR Research Contract N00014-17-1-2177 . An anonymous reviewer made a number of useful comments which improved the narrative of the paper. The authors are grateful to Louisa Thomas for greatly improving the style of presentation.
Publisher Copyright:
© 2020 Elsevier Inc.

PY - 2021/1/15

Y1 - 2021/1/15

N2 - Tutte's spring embedding theorem states that, for a three-connected planar graph, if the outer face of the graph is fixed as the complement of some convex region in the plane, and all other vertices are placed at the mass center of their neighbors, then this results in a unique embedding, and this embedding is planar. It also follows fairly quickly that this embedding minimizes the sum of squared edge lengths, conditional on the embedding of the outer face. However, it is not at all clear how to embed this outer face. We consider the minimization problem of embedding this outer face, up to some normalization, so that the sum of squared edge lengths is minimized. In this work, we show the connection between this optimization problem and the Schur complement of the graph Laplacian with respect to the interior vertices. We prove a number of discrete trace theorems, and, using these new results, show the spectral equivalence of this Schur complement with the boundary Laplacian to the one-half power for a large class of graphs. Using this result, we give theoretical guarantees for this optimization problem, which motivates an algorithm to embed the outer face of a spring embedding.

AB - Tutte's spring embedding theorem states that, for a three-connected planar graph, if the outer face of the graph is fixed as the complement of some convex region in the plane, and all other vertices are placed at the mass center of their neighbors, then this results in a unique embedding, and this embedding is planar. It also follows fairly quickly that this embedding minimizes the sum of squared edge lengths, conditional on the embedding of the outer face. However, it is not at all clear how to embed this outer face. We consider the minimization problem of embedding this outer face, up to some normalization, so that the sum of squared edge lengths is minimized. In this work, we show the connection between this optimization problem and the Schur complement of the graph Laplacian with respect to the interior vertices. We prove a number of discrete trace theorems, and, using these new results, show the spectral equivalence of this Schur complement with the boundary Laplacian to the one-half power for a large class of graphs. Using this result, we give theoretical guarantees for this optimization problem, which motivates an algorithm to embed the outer face of a spring embedding.

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U2 - 10.1016/j.laa.2020.08.035

DO - 10.1016/j.laa.2020.08.035

M3 - Article

AN - SCOPUS:85090333485

SN - 0024-3795

VL - 609

SP - 73

EP - 107

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -