Graph Laplacians do not generate strongly continuous semigroups

Abstract

We show that for graph Laplacians $\Delta_G$ on a connected locally finite simplicial undirected graph $G$ with countable infinite vertex set $V$ none of the operators $\alpha\,\mathrm{Id}+\beta\Delta_G, \alpha,\beta\in\mathbb{K},\beta \ne 0$, generate a strongly continuous semigroup on $\mathbb{K}^V$ when the latter is equipped with the product topology.