# Cut-norm and entropy minimization over weak* limits

## TL;DR

This paper establishes a link between graph limits in cut-distance and weak* limits of adjacency matrices that minimize entropy or similar concave functionals, without relying on the regularity lemma.

## Contribution

It proves that cut-distance accumulation points correspond to entropy-minimizing weak* limit points, providing an elementary proof that avoids the regularity lemma.

## Key findings

- Characterization of cut-norm limits via entropy minimization
- Extension to any strictly concave functional of the adjacency matrix
- Elementary proofs avoiding the regularity lemma

## Abstract

We prove that the accumulation points of a sequence of graphs $G_1,G_2,G_3,\ldots$ with respect to the cut-distance are exactly the weak$^*$ limit points of subsequences of the adjacency matrices (when all possible orders of the vertices are considered) that minimize the entropy over all weak$^*$ limit points of the corresponding subsequence. In fact, the entropy can be replaced by any map $W\mapsto \int\int f(W(x,y))$, where $f$ is a continuous and strictly concave function. Our proofs are elementary, and do not use the regularity lemma.