On Some Properties of the Star Graph

We derive some properties of the star graph in this paper. In particular, we compute the number of nodes at distance i from a fixed node e in a star graph. To this end, a recursive formula is first obtained. This recursive formula is, in general, hard to solve for a closed form solution. We then study the relations among the number of nodes at distance i to node e in star graphs of different dimensions. This study reveals a very interesting relation among these numbers, which leads to a simple homogeneous linear recursive formula whose characteristic equation is easy to solve. Thus, we get a systematic way to obtain a closed form solution with given initial conditions for any fixed i.