Quasirandom Graphs and the Pantograph Equation

TL;DR

This paper reveals that the set of all cliques does not enforce quasirandomness in graphs, using the pantograph equation and deformed exponential functions to connect differential equations with graph theory.

Contribution

It introduces a novel application of the pantograph equation in graph theory, demonstrating that the set of all cliques is not a forcing family for quasirandomness.

Findings

The set of all cliques is not forcing for quasirandomness.

Provides an example of an infinite non-forcing family of graphs.

Answers a question posed by P. Horn regarding graph forcing families.

Abstract

The pantograph differential equation and its solution, the deformed exponential function, are remarkable objects that appear in areas as diverse as combinatorics, number theory, statistical mechanics, and electrical engineering. In this article we describe a new surprising application of these objects in graph theory, by showing that the set of all cliques is not forcing for quasirandomness. This provides a natural example of an infinite family of graphs, which is not forcing, and answers a natural question posed by P. Horn.