Entropy and Enumeration of Subtrees in a Cactus Network - Dong, Lixin; Zhao, Haixing; Lai, Hong-Jian (Frontiers in Physics, 2020)
For a given network, the number of spanning trees is a key parameter to measure its reliability in edge failure cases, while the number of subtrees is a key parameter to measure its reliability in both vertex and edge failures cases. Zhang et al. investigated the entropy of spanning trees, spanning forests and connected spanning subgraphs of Koch networks. In this paper, we extend Koch networks to 3-cactus networks, and study the entropy of subtrees of 3-cactus networks. We present a linear algorithm to count the number of subtrees in a 3-cactus network, determine the upper and lower bounds of the entropy of subtrees of these networks and characterize those attaining the extremal values. As an application, a linear algorithm is developed to count the number of subtrees in Koch networks, with complexity O(g), where g is the number of iterations. Finally, we determine the entropy of subtrees of Koch networks.