Graph Laplacians do not generate strongly continuous semigroups

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

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