Abstract
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.
Highlights
We derive some properties of the star graph in this paper
The vertex symmetry of the star graph implies that routing between two arbitrary nodes reduces to routing from an arbitrary node to the identity node e 123 n
We derived some properties of the star graph
Summary
Et Vn be the set of all n! permutations of symbols 1, 2 n. The problem of computing the number of nodes at a given distance from an origin is studied for rotator graphs in [5]. We continue our study, using a different approach, to obtain a nice recursive formula for the total number of nodes at distance from e in S. For each node of the form 1 k at distance to e, by the greedy algorithm, one optimal path is as follows: Theorem 2 In S,, the total number of nodes at distance from e can be computed recursively as follows: For n >- 1 and
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