# A New Hierarchical Structure of Star Graphs and Applications

## Abstract

A star graph *S* _{ n } [1], of order *n*, is defined to be a symmetric graph *G* = (*V*, *E*) where *V* is the set of *n*! vertices, each representing a distinct permutation of *n* elements and *E* is the set of symmetric edges such that two permutations (nodes) are connected by an edge iff one can be reached from the other by interchanging its first symbol with any other symbol. The star graph *S* _{ n } is a (*n* − 1)-regular graph with *n*! nodes and *n*!(*n* − 1)/2 edges. Recursive hierarchical structure is one of the most attractive and well known properties of star graphs. A dimension *n* star graph can be divided into *n* substars of dimension *n* − 1 by grouping the nodes with the same symbol at the *i*th position together, 2 ≤ *i* ≤ *n* [see Figure 1].

In this paper, we propose a new recursive hierarchical structure of star graphs. The objective is to redesign shortest routing in star graphs in the light of this new structure and design new efficient algorithms for shortest path multicast algorithms [2] adaptibe to bandwidth and latency requirements.

## Keywords

IEEE Transaction Hierarchical Structure Regular Graph Star Graph Latency Requirement## References

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