The star graph, though an attractive alternative to the hypercube, has a major drawback in that the number of nodes for an n-star graph must be n!, and thus considerably limits the choice of the number of nodes in the graph. In order to alleviate this drawback, the arrangement graph was recently proposed as a generalization of the star graph topology. The arrangement graph provides more flexibility than the star graph in choosing the number of nodes, but the degree of the resulting network may be very high. To overcome that disadvantage, this paper presents another generalization of the star graph, called the (n,k)-star graph. We examine some topological properties of the (n,k)-star graph from the graph-theory point of view. It is shown that two different types of edges in the (n,k)-star prevent it from being edge-symmetric, but edges in each class are essentially symmetric with respect to each other. Also, the diameter and the exact average distance of the (n,k)-star graph are derived. In addition, the fault-diameter for the (n,k)-star graph is shown to be at most the fault-free diameter plus three.
Cyclic Decomposition of $k$-Permutations and Eigenvalues of the Arrangement Graphs
