We consider the so-called emph{discrete $p$-Laplacian}, a nonlinear difference operator that acts on functions defined on the nodes of a possibly infinite graph. We study the associated nonlinear Cauchy problem and identify the generator of the assoc
iated nonlinear semigroups. We prove higher order time regularity of the solutions. We investigate the long-time behavior of the solutions and discuss in particular finite extinction time and conservation of mass. Namely, on one hand, for small $p$ if an infinite graph satisfies some isoperimetric inequality, then the solution to the parabolic $p$-Laplace equation vanishes in finite time; on the other hand, for large $p,$ these parabolic $p$-Laplace equations always enjoy conservation of mass.
We present a Gershgorins type result on the localisation of the spectrum of a matrix. Our method is elementary and relies upon the method of Schur complements, furthermore it outperforms the one based on the Cassini ovals of Ostrovski and Brauer. Fur
thermore, it yields estimates that hold without major differences in the cases of both scalar and operator matrices. Several refinements of known results are obtained.
We introduce quantum hypergraphs, in analogy with the theory of quantum graphs developed over the last 15 years by many authors. We emphasize some problems that arise when one tries to define a Laplacian on a hypergraph.
We discuss the Krein--von Neumann extensions of three Laplacian-type operators -- on discrete graphs, quantum graphs, and domains. In passing we present a class of one-dimensional elliptic operators such that for any $nin mathbb N$ infinitely many el
ements of the class have $n$-dimensional null space.
We survey some known results about operator semigroup generated by operator matrices with diagonal or coupled domain. These abstract results are applied to the characterization of well-/ill-posedness for a class of evolution equations with dynamic bo
undary conditions on domains or metric graphs. In particular, our ill-posedness results on the heat equation with general Wentzell-type boundary conditions complement those previously obtained by, among others, Bandle-von Below-Reichel and Vitillaro-Vazquez.
