# From Planck area to graph theory: Topologically distinct black hole microstates

## TL;DR

This paper models black hole microstates using topologically distinct tilings of the horizon, linking geometric configurations to graph theory, and derives the entropy spectrum consistent with the Bekenstein-Hawking formula.

## Contribution

It introduces a novel topological and graph-theoretic framework for counting black hole microstates, providing an exact entropy spectrum in 2+1 dimensions.

## Key findings

- Entropy formula matches Bekenstein-Hawking with logarithmic correction
- Enumeration of tilings yields a precise microstate count
- Provides a graph-theoretic interpretation of horizon microstates

## Abstract

We postulate a Planck scale horizon unit area, with no bits of information locally attached to it, connected but otherwise of free form, and let $n$ such geometric units compactly tile the black hole horizon. Associated with each topologically distinct tiling configuration is then a simple, connected, undirected, unlabeled, planar, chordal graph. The asymptotic enumeration of the corresponding integer sequence gives rise to the Bekenstein-Hawking area entropy formula, automatically accompanied by a proper logarithmic term, and fixes the size of the horizon unit area, thereby constituting a global realization of Wheeler's "it from bit" phrase. Invoking Polya's theorem, an exact number theoretical entropy spectrum is offered for the 2+1 dimensional quantum black hole.