Counting non-isomorphic maximal independent sets of the n-cycle graph
Abstract
The number of maximal independent sets of the n-cycle graph C_n is known to be the nth term of the Perrin sequence. The action of the automorphism group of C_n on the family of these maximal independent sets partitions this family into disjoint orbits, which represent the non-isomorphic (i.e., defined up to a rotation and a reflection) maximal independent sets. We provide exact formulas for the total number of orbits and the number of orbits having a given number of isomorphic representatives. We also provide exact formulas for the total number of unlabeled (i.e., defined up to a rotation) maximal independent sets and the number of unlabeled maximal independent sets having a given number of isomorphic representatives. It turns out that these formulas involve both Perrin and Padovan sequences.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- January 2007
- DOI:
- 10.48550/arXiv.math/0701647
- arXiv:
- arXiv:math/0701647
- Bibcode:
- 2007math......1647B
- Keywords:
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- Mathematics - Combinatorics;
- Computer Science - Discrete Mathematics;
- Mathematics - Group Theory;
- 05C69;
- 05C38;
- 05C25 (Primary) 05A15;
- 05A17;
- 11Y55 (Secondary)
- E-Print:
- Revised version