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Description

The $n$th eigenfunction of the Laplacian (-d^2/dx^2) on a quantum graph divides the graph into at most $n$ regions (known as nodal domains). We study the relationship between the $n$th eigenfunction of the Laplacian and partitions of the graph into $n$ parts. We describe a procedure of attaining eigenfunctions, and hence the spectrum, by investigating these partitions, in particular minimal ones. The minimal partitions are found partially by assigning a score to each partition which is the maximum of the first eigenvalue of the Laplacian over each part of the partition. No prior knowledge of quantum graphs is necessary. This work is based partially on results by Bernard Helffer, Thomas Hoffmann-Ostenhoff, Susanna Terracini et al. Joint work with Rami Band, Gregory Berkolaiko and Uzy Smilansky