The number of subtrees, or simply the subtree number, is one of the most studied counting-based graph invariants that has applications in many interdisciplinary fields such as phylogenetic reconstruction. Motivated from the study of graph surgeries on evolutionary dynamics, we consider the subtree problems of fan graphs, wheel graphs, and the class of graphs obtained from “partitioning” wheel graphs under dynamic evolution. The enumeration of these subtree numbers is done through the so-called subtree generation functions of graphs. With the enumerative result, we briefly explore the extremal problems in the corresponding class of graphs. Some interesting observations on the behavior of the subtree number are also presented.

With generating function and structural analysis, this paper presents the subtree generating functions and the subtree number index of generalized book graphs, generalized fan graphs, and generalized wheel graphs, respectively. As an application, this paper also briefly studies the subtree number index and the asymptotic properties of the subtree densities in regular book graphs, regular fan graphs, and regular wheel graphs. The results provide the basis for studying novel structural properties of the graphs generated by generalized book graphs, fan graphs, and wheel graphs from the perspective of the subtree number index.

The subtree number index of a graph, defined as the number of subtrees, attracts much attention recently. Finding a proper algorithm to compute this index is an important but difficult problem for a general graph. Even for unicyclic and bicyclic graphs, it is not completely trivial, though it can be figured out by try and error. However, it is complicated for tricyclic graphs. This paper proposes path contraction carrying weights (PCCWs) algorithms to compute the subtree number index for the nontrivial case of bicyclic graphs and all 15 cases of tricyclic graphs, based on three techniques: PCCWs, generating function and structural decomposition. Our approach provides a foundation and useful methods to compute subtree number index for graphs with more complicated cycle structures and can be applied to investigate the novel structural property of some important nanomaterials such as the pentagonal carbon nanocone.

<p>Link prediction is one of the most important tasks in uncovering evolving mechanisms of dynamic complex networks. Existing dynamic link prediction models suffer from limitations such as vulnerability to adversarial attacks, poor accuracy, and instability. In this paper, we propose a novel dynamic Graph Convolutional Network model incorporating a Self-adaptive Stable Gate (SAGE-GCN) consisting of a state encoding network and a policy network. Firstly, we capture the local topology of the nodes by employing a multi-power adjacency matrix to obtain higher-order topological features, enabling its features to be distinguished at different network snapshots. Then, a stable gate is introduced to ensure multiple spatiotemporal dependency paths within the state encoding network. It is proven that SAGE-GCN with integral Lipschitz graph convolution is stable to relative perturbations in the dynamic networks. Finally, a self-adaptive strategy is proposed to choose different state encoding network instances, with a policy network used to learn the optimal temporal and structural features through corresponding rewards to capture network dynamics. With the aid of extensive experiments on five real-world graph benchmarks, SAGE-GCN is shown to substantially outperform current state-of-the-art approaches in terms of precision and stability of dynamic link prediction and ability to successfully defend against various attacks.</p>

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