Some topological properties of star graphs: The surface area and volume

The star graph, as an interesting network topology, has been extensively studied in the past. In this paper, we address some of the combinatorial properties of the star graph. In particular, we consider the problem of calculating the surface area and volume of the star graph, and thus answering an open problem previously posed in the literature. The surface area of a sphere with radius ii in a graph is the number of nodes in the graph whose distance from a given node is exactly ii. The volume of a sphere with radius ii in a graph is the number of nodes within distance ii from the given node. In this paper, we derive explicit expressions to calculate the surface area and volume in the star graph.