The tau constant and the discrete Laplacian matrix of a metrized graph

TL;DR

This paper establishes a relationship between the tau constant of a metrized graph and the discrete Laplacian matrix, providing a new way to compute the tau constant using matrix methods.

Contribution

It introduces a novel expression of the tau constant in terms of the discrete Laplacian matrix and its pseudo inverse, linking continuous and discrete graph invariants.

Findings

Tau constant expressed via discrete Laplacian matrix

Provides a computational method for tau constant

Bridges continuous and discrete graph invariants

Abstract

We express the tau constant of a metrized graph in terms of the discrete Laplacian matrix and its pseudo inverse.