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We prove well-posedness for very general linear wave- and diffusion equationson compact or non-compact metric graphs allowing various different conditionsin the vertices. More precisely, using the theory of strongly continuousoperator semigroups we show that a large class of (not necessarilyself-adjoint) second order differential operators with general (possiblynon-local) boundary conditions generate cosine families, hence also analyticsemigroups, on${\mathrm{L}}^p({\mathbb{R}_+},{\mathbb{C}}^{\ell})\times{\mathrm{L}}^p([0,1],{\mathbb{C}}^m)$for $1\le p<+\infty$.

Waves and Diffusion on Metric Graphs with General Vertex Conditions