Spanning trees in a closed chain of planar networks

The enumeration of spanning trees in a finite graph is an important problem related to various domains of mathematics, physics and network reliability that has been investigated by many researchers. A network N is called a closed chain of planar networks if its modelling is defined by n planar graphs connected by articulation points. Some recent studies focus on finding the number of spanning trees in the star flower network which is a closed chain of cycles. In this paper, we give a general method counting the number of spanning trees in a closed chain of planar networks. We propose a recursive approach based on the contraction of nodes and as application, we obtain explicit expressions for the number of spanning trees in the closed chain of maximal planar networks and the pseudofractal scale-free network.