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Schrodinger operators, operators on manifolds, general differential operators, numerical studies, integral operators, discrete models, resonances, non-self-adjoint operators, random operators/matrices
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Decomposition of a regular graph and connected subgraphs
Edit: How about this counterexample for the case $s_2,t_2>0$:
$G=(V,E)$ with $V=\{x_1\}\cup\{u_i^k\ |\ 1\le i\le n,1\le k\le 4\}$, and $E$ being the union of
$\{(x_1,u_i^k)\ |\ 1\le i\le n, k\in\{1 …