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We prove an Ohsawa-Takegoshi-type extension theorem on the Berkovich closed unit disc over certain non-Archimedean fields. As an application, we establish a non-Archimedean analogue of Demailly's regularization theorem for quasisubharmonic functions on the Berkovich unit disc.Date: May 9, 2018.

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Let G be a finite connected simple graph. We define the moduli space of conformal structures on G. We propose a definition of conformally covariant operators on graphs, motivated by [25]. We provide examples of conformally covariant operators, which include the edge Laplacian and the adjacency matrix on graphs. In the case where such an operator has a nontrivial kernel, we construct conformal invariants, providing discrete counterparts of several results in [11,12] established for Riemannian manifolds. In particular, we show that the nodal sets and nodal domains of null eigenvectors are conformal invariants. Proposition 6.9. For each (i 1 , i 2 ) ∈ I ×I, Λ i1,i2 (J, F, w) is conformally covariant.Proof. Let D i1 J (u) be the matrix obtained from D J (u) by removing the i 1 st row and i 1 st column, and let D i2 J (u) be the matrix obtained from D J (u) by removing the i 2 nd row and i 2 nd column. Then, it follows from Lemma 6.8 that Λ i1,i2 (J, F,w) = D i1 J (u) · Λ i1,i2 (J, F, w) · D i2 J (u).