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

TL;DR

This paper uncovers new explicit relationships between discrete and continuous operators on metric graphs, linking the Laplacian, averaging operator, and wave equation solutions, applicable to infinite graphs.

Contribution

It introduces novel explicit formulas connecting discrete and continuous transition operators on metric graphs, expanding analysis to infinite graphs without local finiteness assumptions.

Findings

Operators can be expressed through each other explicitly

Averaging operator relates closely to wave equation solutions

Applicable to infinite graphs without local finiteness

Abstract

We establish several new relations between the discrete transition operator, the continuous Laplacian and the averaging operator associated with combinatorial and metric graphs. It is shown that these operators 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 allows one to study a class of infinite graphs without assumption on the local finiteness.