It is relatively easy to compute the number of nodal domains for a given eigenfunction1. Thus it is no problem to compute the possible number of nodal domains when all eigenvalues are simple. The situation changes completely in the case of degenerate eigenvalues because then the number of nodal domains may vary considerably depending on which eigenfunction from the r-dimensional eigenspace of λk is chosen. Hence, given a fixed graph G(V,E) and an eigenvalue λk of multiplicity r three questions immediately arise.
KeywordsLocal Optimum Computational Experiment Gaussian Random Variable Hyperplane Arrangement Monte Carlo Integration
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