# Dynamical systems associated with adjacency matrices

## TL;DR

This paper develops the theory of linear evolution equations linked to adjacency matrices of graphs, analyzing their qualitative properties and well-posedness, especially on infinite and line graphs, with applications to cycle detection.

## Contribution

It introduces a new framework for studying evolution equations on graphs, distinguishing between backward and forward types, and explores their properties and applications.

## Key findings

- Forward equations exhibit diffusive features but may be ill-posed on graphs with unbounded degree.
- Backward equations are well-posed on line graphs.
- Cycle detection can be achieved through backward equations on line graphs.

## Abstract

We develop the theory of linear evolution equations associated with the adjacency matrix of a graph, focusing in particular on infinite graphs of two kinds: uniformly locally finite graphs as well as locally finite line graphs. We discuss in detail qualitative properties of solutions to these problems by quadratic form methods. We distinguish between backward and forward evolution equations: the latter have typical features of diffusive processes, but cannot be well-posed on graphs with unbounded degree. On the contrary, well-posedness of backward equations is a typical feature of line graphs. We suggest how to detect even cycles and/or couples of odd cycles on graphs by studying backward equations for the adjacency matrix on their line graph.