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A set of spanning trees in a graph G is called independent spanning trees (ISTs) if they are rooted at the same vertex r, and for each vertex v( = r) in G, the two paths from v to r in any two trees share no common vertex expect for v and r. ISTs can be applied in many research fields, such as fault-tolerant broadcasting and secure message distribution in reliable communication networks. Since Cayley graphs have been widely used to design interconnection networks, constructing ISTs on cayley graphs is worth studying. The alternating group network is a subclass of Cayley graphs, and the approach of constructing ISTs in alternating group networks is still unknown. In this paper, we propose a novel and simple top-down approach for constructing ISTs in alternating group networks. The main ideas of the algorithm are to use induction to develop small trees to big trees, to use a triangle breadth-first search (TBFS) process to create a backbone of an IST, and to use breadth-first search (BFS) process to connect the rest of nodes. Compared to other methods of different interconnection networks in the literature, the uniqueness of our method is that it does not need to determine the parent of one node by any rule, on the contrary others determine that by rules. The time complexity in n-dimensional alternating group network AN n is O(n 2 × n!), where n! is twice the number of nodes of AN n ; hence it is polynomial time. We implement the algorithm in PHP and run cases from AN 3 to AN 10 . The results reveal that all spanning trees of AN 3 to AN 10 are independent and that our algorithm is accurate and efficient.

Hub location problems originated in telecommunications and transportation systems. They combine various aspects of network issues, including location problems, network design problems and routing problems. The main theme of these problems is to route the services via a subset of hubs, rather than route each service with direct links between demand nodes. Therefore, solutions for hub location problems typically use sets of hubs to reroute the flow of the services and attempt to reduce the costs of building the service links. In this paper, we present a survey of hub location problems.

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The maximum k-plex problem is intended to relax the clique definition with maximum cardinality. It helps people understand the structures of networks by considering them a problem of graph clustering; therefore, it is a useful model for social network analysis. In this paper, we propose a swap-based heuristic algorithm for the maximum k-plex problem. Starting with a randomly selected solution, we use three types of swap operators to enlarge our current solution in different situations. The results of experiments conducted on various graphs from two benchmarks demonstrated that our proposed algorithm achieved desirable performance on specific graphs compared with other algorithms.