# Lipschitz property for systems of linear mappings and bilinear forms

## TL;DR

This paper establishes a Lipschitz continuity property for the isomorphism classes of representations of graphs with both undirected and directed edges, linking small perturbations to small changes in the isomorphism.

## Contribution

It proves that for such graph representations, isomorphisms can be chosen to vary continuously with the representations, providing a Lipschitz-type stability result.

## Key findings

- Isomorphic representations close to each other admit near-identity isomorphisms.
- The result applies to systems with bilinear forms and linear maps on graph edges.
- Provides a stability guarantee for graph representation isomorphisms.

## Abstract

Let G be a graph with undirected and directed edges. Its representation is given by assigning a vector space to each vertex, a bilinear form on the corresponding vector spaces to each directed edge, and a linear map to each directed edge. Two representations A and A' of G are called isomorphic if there is a system of linear bijections between the vector spaces corresponding to the same vertices that transforms A to A'. We prove that if two representations are isomorphic and close to each other, then their isomorphism can be chosen close to the identity.