Geometric approach to graph magnitude homology

TL;DR

This paper introduces a geometric method for computing graph magnitude homology, providing new proofs for trees and extending to graphs with cycles, with demonstrated computations.

Contribution

It presents a novel geometric approach to compute magnitude homology, including a new proof for trees and methods applicable to general graphs with cycles.

Findings

Trees are diagonal in magnitude homology.

The new method simplifies computations for graphs with cycles.

Demonstrated computations validate the approach.

Abstract

In this paper, we introduce a new method to compute magnitude homology of general graphs. To each direct sum component of magnitude chain complexes, we assign a pair of simplicial complexes whose simplicial chain complex is isomorphic to it. First we states our main theorem specialized to trees, which gives another proof for the known fact that trees are diagonal. After that, we consider general graphs, which may have cycles. We also demonstrate some computation as an application.