# Preorder Construct on Simple Undirected Graphs

## TL;DR

This paper introduces a new preorder on graph nodes that aligns with the coarsest equitable partition, providing invariants for dynamical systems like the logistic model on graphs.

## Contribution

It constructs a novel preorder related to graph topology that preserves the coarsest equitable partition and offers a functional characterization alternative.

## Key findings

- Preorder is preserved by logistic dynamical systems.
- Equivalence relation of the preorder matches the coarsest equitable partition.
- Provides a new invariant set for network dynamical systems.

## Abstract

We construct a novel preorder on the set of nodes of a simple undirected graph. We prove that the preorder (induced by the topology of the graph) is preserved, e.g., by the logistic dynamical system (both in discrete and continuous time). Moreover, the underlying equivalence relation of the preorder corresponds to the coarsest equitable partition (CEP). This will further imply that the logistic dynamical system on a graph preserves its coarsest equitable partition. The results provide a nontrivial invariant set for the logistic and the like dynamical systems, as we show. We note that our construct provides a functional characterization for the CEP as an alternative to the pure set theoretical iterated degree sequences characterization. The construct and results presented might have independent interest for analysis on graphs or qualitative analysis of dynamical systems over networks.