This paper addresses the second-order consensus problem for multiagent systems with nonlinear dynamics and directed topologies. The authors define a new concept called generalized algebraic connectivity for strongly connected networks and extend it to the strongly connected components of directed networks containing a spanning tree. They derive sufficient conditions for achieving second-order consensus based on algebraic graph theory, matrix theory, and Lyapunov control approaches. The paper includes numerical examples to validate the theoretical analysis. Key findings include the importance of the generalized algebraic connectivity in determining the ability to reach consensus and the conditions under which second-order consensus can be achieved in both strongly connected networks and networks with directed spanning trees.This paper addresses the second-order consensus problem for multiagent systems with nonlinear dynamics and directed topologies. The authors define a new concept called generalized algebraic connectivity for strongly connected networks and extend it to the strongly connected components of directed networks containing a spanning tree. They derive sufficient conditions for achieving second-order consensus based on algebraic graph theory, matrix theory, and Lyapunov control approaches. The paper includes numerical examples to validate the theoretical analysis. Key findings include the importance of the generalized algebraic connectivity in determining the ability to reach consensus and the conditions under which second-order consensus can be achieved in both strongly connected networks and networks with directed spanning trees.