Enumeration of spanning trees in planar unclustered networks

Among a variety of subgraphs, spanning trees are one of the most important and fundamental categories. They are relevant to diverse aspects of networks, including reliability, transport, self-organized criticality, loop-erased random walks and so on. In this paper, we introduce a family of modular, self-similar planar networks with zero clustering. Relevant properties of this family are comparable to those networks associated with technological systems having low clustering, like power grids, some electronic circuits, the Internet and some biological systems. So, it is very significant to research on spanning trees of planar networks. However, for a large network, evaluating the relevant determinant is intractable. In this paper, we propose a fairly generic linear algorithm for counting the number of spanning trees of a planar network. Using the algorithm, we derive analytically the exact numbers of spanning trees in planar networks. Our result shows that the computational complexity is O(t), which is better than that of the matrix tree theorem with O(m2t2), where t is the number of steps and m is the girth of the planar network. We also obtain the entropy for the spanning trees of a given planar network. We find that the entropy of spanning trees in the studied network is small, which is in sharp contrast to the previous result for planar networks with the same average degree. We also determine an upper bound and a lower bound for the numbers of spanning trees in the family of planar networks by the algorithm. As another application of the algorithm, we give a formula for the number of spanning trees in an outerplanar network with small-world features.