Optimal inter-layer connections for maximizing synchronizability in two-layer chain network.

We develop a rigorous mathematical framework to determine the optimal inter-layer edge configurations that maximize synchronizability in two-layer chain networks-an area previously limited to empirical approaches. Departing from prior work relying on numerical simulations, we analytically prove that synchronizability is maximized when inter-layer edges are placed (i) at the chain's midpoint (single-edge case) and (ii) at the one-quarter and three-quarter positions (dual-edge case). We also compute the coupling strength thresholds and further conjecture the optimal placement pattern for an arbitrary number of inter-layer edges, supporting this hypothesis with extensive numerical validation. These results bridge spectral graph theory and network topology design, offering principled guidelines for engineering interconnected chain-like systems, such as supply chains, information dissemination, and epidemic spread.