A connected graph G is called strongly Menger edge connected if G has min{deg G (x), deg G (y)} edge-disjoint paths between any two distinct vertices x and y in G. In this paper, we consider two types of strongly Menger edge connectivity of the line graphs of n-dimensional hypercube-like networks with faulty edges, namely the m-edgefault-tolerant and m-conditional edge-fault-tolerant strongly Menger edge connectivity. We show that the line graph of any n-dimensional hypercube-like network is (2n − 4)edge-fault-tolerant strongly Menger edge connected for n ≥ 3 and (4n − 10)-conditional edge-fault-tolerant strongly Menger edge connected for n ≥ 4. The two bounds for the maximum number of faulty edges are best possible.

Small-world networks are ubiquitous in real-life systems, such as the World Wide Web, communication networks, and electric power grids, and most of them are stochastic. In this paper, we present a model that generates a small-world network in a simple deterministic way and analyze the relevant topological properties of the model, such as the degree distribution, clustering coefficient, and diameter. Meanwhile, according to the special structure of the model, we derive analytically the exact numbers of spanning trees in the planar networks. The results show that the model has a discrete exponential degree distribution, high clustering coefficient, short diameter, and high entropy.

Complex networks have seen much interest from all research fields and have found many potential applications in a variety of areas including natural, social, biological, and engineering technology. The deterministic models for complex networks play an indispensable role in the field of network model. The construction of a network model in a deterministic way not only has important theoretical significance, but also has potential application value. In this paper, we present a class of 3-regular network model with small world phenomenon. We determine its relevant topological characteristics, such as diameter and clustering coefficient. We also give a calculation method of number of spanning trees in the 3-regular network and derive the number and entropy of spanning trees, respectively.