This paper investigates the leader-following consensus problem in multi-agent systems with time-varying coupling delays. Two cases of coupling topologies are considered: fixed and switched. For the fixed topology case, a necessary and sufficient condition is derived for consensus, while for the switched topology case, a sufficient condition is proposed. The analysis uses Lyapunov-Razumikhin functions and linear matrix inequalities. Numerical examples are provided to illustrate the results. The study shows that the convergence of the system depends on the global reachability of the leader node in the graph. When the coupling topology is fixed and directed, the leader node must be globally reachable for consensus. When the topology is switched and balanced, a sufficient condition is given for convergence. The results are validated through simulations with different coupling topologies and time delays. The paper concludes that the leader-following consensus can be achieved under certain conditions, even with time delays and switching topologies.This paper investigates the leader-following consensus problem in multi-agent systems with time-varying coupling delays. Two cases of coupling topologies are considered: fixed and switched. For the fixed topology case, a necessary and sufficient condition is derived for consensus, while for the switched topology case, a sufficient condition is proposed. The analysis uses Lyapunov-Razumikhin functions and linear matrix inequalities. Numerical examples are provided to illustrate the results. The study shows that the convergence of the system depends on the global reachability of the leader node in the graph. When the coupling topology is fixed and directed, the leader node must be globally reachable for consensus. When the topology is switched and balanced, a sufficient condition is given for convergence. The results are validated through simulations with different coupling topologies and time delays. The paper concludes that the leader-following consensus can be achieved under certain conditions, even with time delays and switching topologies.