The spectral graph theory is introduced to establish the effective resistance matrix of power system network topology from the mathematical point of view, and the index of nodal coupling degree is put forward based on the effective resistance matrix. By analyzing the cumulative probability distribution of node coupling degree, the scale-free characteristic of power system is verified. Then the index is applied to identification of the key nodes in actual power system. The index not only satisfies the Kirchhoff's law, but also can show the quantitative relationship between nodal phase angle and nodal power, meanwhile, its physical meaning is more consistent with actual power system. In addition, the capacity survivability index, load survivability index, line survivability index and synthetical survivability index are adopted to describe the effects of failure of the nodes on power system. Finally, the simulation verification of IEEE-39 bus system is done, which shows that the index of nodal coupling degree has a certain recognition to the nodes in the key position of the system topology, so it can provide guidance for the safe and stable operation of actual power system. This work is supported by Power System Planning Subject of Power Dispatch and Control Center of Guangdong Power Grid Co., Ltd. (No. 036000QQ00150002).