# Remarks on BEC on graphs

## TL;DR

This paper investigates Bose--Einstein condensation on graphs with transient adjacency matrices, revealing a non-factor quasi-free state exhibiting BEC that decomposes into generalized coherent states, and reviews conditions for their properties.

## Contribution

It introduces a non-factor quasi-free state exhibiting BEC on graphs and analyzes the properties of generalized coherent states in this context.

## Key findings

- The BEC state is non-factor and decomposes into generalized coherent states.
- Conditions for generalized coherent states to be faithful, factor, and pure are reviewed.
- Generalized coherent states are shown to be quasi-equivalent under certain conditions.

## Abstract

We consider Bose--Einstein condensation (BEC) on graphs with transient adjacency matrix and obtain a quasi-free state exhibiting BEC is non-factor and decompose into generalized coherent states. We review necessary and sufficient conditions that a generalized coherent state is faithful, factor, and pure and generalized coherent states are quasi-equivalent as well.