From weighted to unweighted graphs in Synchronizing Graph Theory

Abstract

A way to associate unweighted graphs from weighted ones is presented, such that linear stable equilibria of the Kuramoto homogeneous model associated to both graphs coincide, i.e., equilibria of the system ˙ θ i = ∑ j ∼ i sin ( θ j − θ j ) , where i ∼ j means vertices i and j are adjacent in the corresponding graph. As a consequence, the existence of linearly stable equilibrium is proved to be NP-Hard as conjectured by R. Taylor in 2015 and a new lower bound for the minimum degree that ensures synchronization is found.