Synchronizing Graphs

• Jan Ahmed, Mathematics, University of Delaware

• Sebastian Cioaba, Mathematics, University of Delaware

In the year 1665, Christian Huygens observed that two pendulum clocks hung from the same board synchronized. This observation leads to the more general problem of having a network of coupled oscillators. In 1975, Yoshiki Kuramoto found a model, called the Kuramoto model, that describes the behavior of a network of coupled oscillators over time. The natural occurrence of the model is found in fireflies synchronizing at night, neurons synchronizing in the brain, and other systems of chemical and biological oscillators. Researchers studying the Kuramoto model showed that for specific networks with certain initial phases of every oscillator, the model does not synchronize. This motivated the question of determining the types of networks that synchronize no matter the initial phases of each oscillator. We call these networks globally synchronizing graphs. Some of the trivial families of graphs have already been studied and it was shown that for complete graphs, trees and cycles of length less than or equal to four are globally synchronizing. It was also shown that graphs that have every vertex connected to at least 78.89 percent of all other vertices are globally synchronizing. In 2022, it was shown that expander graphs satisfying specific bounds globally synchronize. Our current aim is to look at other families of graphs and determine whether or not they globally synchronize.