New Relations Between Discrete and Continuous Transition Operators on (Metric) Graphs

We establish several new relations between the discrete transition operator, the continuous Laplacian and the averaging operator associ-ated with combinatorial and metric graphs. It is shown that these oper-ators can be expressed through each other using explicit expressions. In particular, we show that the averaging operator is closely related with the solutions of the associated wave equation. The machinery used al-lows one to study a class of infinite graphs without assumption on the local finiteness.