# Exactly Solvable Random Graph Ensemble with Extensively Many Short Cycles

## TL;DR

This paper introduces a new ensemble of 2-regular random graphs with tunable short cycle distributions, analyzing their phase transitions, spectral properties, and equivalence of different statistical formulations.

## Contribution

It presents an exactly solvable model of random graphs with many short cycles, including a detailed phase diagram and spectral density analysis.

## Key findings

- Identifies a second order phase transition between connected and disconnected phases.
- Derives the spectral density with discrete eigenvalues for short cycles.
- Shows equivalence of canonical and grand canonical ensembles in the thermodynamic limit.

## Abstract

We introduce and analyse ensembles of 2-regular random graphs with a tuneable distribution of short cycles. The phenomenology of these graphs depends critically on the scaling of the ensembles' control parameters relative to the number of nodes. A phase diagram is presented, showing a second order phase transition from a connected to a disconnected phase. We study both the canonical formulation, where the size is large but fixed, and the grand canonical formulation, where the size is sampled from a discrete distribution, and show their equivalence in the thermodynamical limit. We also compute analytically the spectral density, which consists of a discrete set of isolated eigenvalues, representing short cycles, and a continuous part, representing cycles of diverging size.