The Number of Large Graphs with a Positive Density of Triangles

TL;DR

This paper establishes bounds on the quantity of fixed-degree graphs with a positive triangle density, revealing their scarcity compared to unrestricted graphs and indicating triangle clustering even at low densities.

Contribution

It provides new upper and lower bounds on the number of fixed-degree graphs with positive triangle density, highlighting their rarity and clustering behavior.

Findings

Few fixed-degree graphs have positive triangle density compared to all graphs.

Triangles tend to cluster even at low densities in these graphs.

The paper offers bounds that quantify the scarcity of such graphs.

Abstract

We give upper and lower bounds on the number of graphs of fixed degree which have a positive density of triangles. In particular, we show that there are very few such graphs, when compared to the number of graphs without this restriction. We also show that in this case the triangles seem to cluster even at low density.