Computation of Polarized Metrized Graph Invariants By Using Discrete Laplacian Matrix

TL;DR

This paper introduces efficient algorithms for computing invariants of polarized metrized graphs using the discrete Laplacian matrix, facilitating both symbolic and numerical calculations in arithmetic geometry.

Contribution

The paper develops novel algorithms that express graph invariants in terms of the discrete Laplacian matrix and its pseudo-inverse, enabling faster computations.

Findings

Algorithms successfully compute invariants for complex graphs.

Implementation examples demonstrate practical efficiency.

Applicable to symbolic and numerical analysis.

Abstract

Several invariants of polarized metrized graphs and their applications in Arithmetic Geometry are studied recently. In this paper, we give fast algorithms to compute these invariants by expressing them in terms of the discrete Laplacian matrix and its pseudo inverse. Algorithms we give can be used for both symbolic and numerical computations. We present various examples to illustrate the implementation of these algorithms.