Benign nonconvexity of synchronization landscape induced by graph skeletons

We study the homogeneous Kuramoto model on a graph and the associated nonconvex optimization problem min_boldsymbolθ∈ℝn-frac12sum1≤ i,j≤ nAijcos(θi-θj). The objective defines an energy over configurations of points on the unit circle and serves as a Lyapunov potential for the dynamics. We prove that every connected quasi-threshold graph is second-order globally synchronizing: every second-order stationary point is a global minimizer corresponding to full synchronization. Consequently, the dynamics converge to full synchronization from almost every initial condition. These graphs are precisely the comparability graphs of partially ordered sets induced by rooted trees. Viewing these trees as graph skeletons, we establish an upward propagation mechanism: synchronization within each child subtree forces the parent and all its descendants to synchronize. The argument proceeds from the leaves to the root and relies on phasor geometry and second-order optimality conditions. Our result provides a structural route to global synchronization that complements existing results based on minimum-degree or spectral proximity to complete graphs.