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Item type:Publication, Access status: without files , Spectral theory for Schrödinger operators on compact metric graphs with δ and δ′ couplings : a surveySpectral properties of Schrödinger operators on compact metric graphs are studied, and special emphasis is put on differences in the spectral behavior between different classes of vertex conditions. We survey recent results especially for δ and δ′ couplings and demonstrate the spectral properties on many examples. Among other things, properties of the ground state eigenvalue and eigenfunction and the spectral behavior under various perturbations of the metric graph or the vertex conditions are considered.Publicationtype: Conference Paper44 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Waiter-Client Games on Randomly Perturbed GraphsWaiter-Client games are played on a hypergraph (X, F), where F⊆ 2X denotes the family of winning sets. During each round, Waiter offers a predefined amount (called bias) of elements from the board X, from which Client takes one for himself while the rest go to Waiter. Waiter wins the game if she can force Client to occupy any winning set F∈ F. In this paper we consider Waiter-Client games played on randomly perturbed graphs. These graphs consist of the union of a deterministic graph Gα on n vertices with minimum degree at least αn and the binomial random graph Gn,p. Depending on the bias we determine the order of the threshold probability for winning the Hamiltonicity game and the k-connectivity game on Gα∪ Gn,p.Publicationtype: Journal Article137