Application of graph theory to the synchronization in an array of coupled nonlinear oscillators

In this letter, we show how algebraic graph theory can be used to derive sufficient conditions for an array of resistively coupled nonlinear oscillators to synchronize. These conditions are derived from the connectivity graph, which describes how the oscillators are connected. In particular, we show how such a sufficient condition is dependent on the algebraic connectivity of the connectivity graph. Intuition tells us that if the oscillators are more "closely connected" to each other, then they are more likely to synchronize. We discuss how to quantify connectedness in graph-theoretical terms and its relation to algebraic connectivity and show that our results are in accordance with this intuition. We also give an upper bound on the coupling conductance required for synchronization for arbitrary graphs, which is in the order of n/sup 2/, where n is the number of oscillators.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>