For generic parameters, the Bloch variety of a nontrivial periodic graph operator is irreducible if and only if the quotient graph is connected.
Kuchment, An overview of periodic elliptic operators , Bulletin of the American Mathemat- ical Society, 53 (2016), p
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Develops and analyzes single- and double-layer potential operators for doubly-periodic harmonic functions on finitely-connected tori, proves compactness and boundary limits, constructs the null space for multiply-connected cases, and demonstrates spectral convergence for Dirichlet, Neumann, and Stek
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Generic Irreducibility of Bloch Varieties for Periodic Graph Operators
For generic parameters, the Bloch variety of a nontrivial periodic graph operator is irreducible if and only if the quotient graph is connected.
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Layer Potential Methods for Doubly-Periodic Harmonic Functions
Develops and analyzes single- and double-layer potential operators for doubly-periodic harmonic functions on finitely-connected tori, proves compactness and boundary limits, constructs the null space for multiply-connected cases, and demonstrates spectral convergence for Dirichlet, Neumann, and Stek