# On the Decomposition of the Laplacian on Metric Graphs

## TL;DR

This paper investigates how the Laplacian operator on symmetric metric graphs can be decomposed into simpler components, revealing structural insights and extending methods from metric trees to more general graphs.

## Contribution

It introduces a decomposition technique for the Laplacian on family preserving metric graphs, generalizing existing methods from metric trees to broader graph structures.

## Key findings

- Decomposition into one-dimensional operators is possible for family preserving metric graphs.
- The structure of the graph directly influences the properties of the decomposed operators.
- A correspondence between discrete and continuum decompositions is established.

## Abstract

We study the Laplacian on family preserving metric graphs. These are graphs that have a certain symmetry that, as we show, allows for a decomposition into a direct sum of one-dimensional operators whose properties are explicitly related to the structure of the graph. Such decompositions have been extremely useful in the study of Schr\"odinger operators on metric trees. We show that the tree structure is not essential, and moreover, obtain a direct and simple correspondence between such decompositions in the discrete and continuum case.