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Quasirandom Graphs and the Pantograph Equation.

Asaf Shapira, Mykhaylo Tyomkyn

The American Mathematical Monthly : the Official Journal of the Mathematical Association of America

|August 30, 2021

Summary

This summary is machine-generated.

The pantograph differential equation and its deformed exponential function have a new application in graph theory. Researchers found that the set of all cliques is not forcing for quasirandomness in graphs.

Keywords:

30D20 MSC: Primary 05C35 Secondary 05C80

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Area of Science:

Graph Theory
Combinatorics
Number Theory
Statistical Mechanics
Electrical Engineering

Background:

The pantograph differential equation and its solution, the deformed exponential function, are significant mathematical objects.
These functions have found applications across diverse scientific fields, including combinatorics, number theory, statistical mechanics, and electrical engineering.

Purpose of the Study:

To explore a novel application of the pantograph differential equation and its solution in graph theory.
To investigate whether the set of all cliques is forcing for quasirandomness in graphs.
To provide a natural example of an infinite family of graphs that is not forcing.

Main Methods:

The study utilizes concepts related to the pantograph differential equation and deformed exponential functions.
Graph theory principles are applied to analyze the properties of graph families.
The research focuses on the concept of 'forcing' in relation to quasirandomness.

Main Results:

A new application of the pantograph differential equation and deformed exponential functions in graph theory is presented.
It is demonstrated that the set of all cliques is not forcing for quasirandomness.
This finding provides a natural example of an infinite family of graphs that is not forcing.

Conclusions:

The research introduces a surprising connection between the pantograph differential equation and graph theory.
The study confirms that the set of all cliques is not forcing for quasirandomness.
This work answers a natural question posed by P. Horn regarding forcing in graph theory.