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We establish several new relations between the discrete transition operator,the continuous Laplacian and the averaging operator associated withcombinatorial and metric graphs. It is shown that these operators can beexpressed through each other using explicit expressions. In particular, we showthat the averaging operator is closely related with the solutions of theassociated wave equation. The machinery used allows one to study a class ofinfinite graphs without assumption on the local finiteness.

New relations between discrete and continuous transition operators on (metric) graphs