# Counting non-isomorphic maximal independent sets of the n-cycle graph

@article{Bisdorff2007CountingNM, title={Counting non-isomorphic maximal independent sets of the n-cycle graph}, author={Raymond Bisdorff and Jean-Luc Marichal}, journal={ArXiv}, year={2007}, volume={abs/math/0701647} }

The number of maximal independent sets of the n-cycle graph Cn is known to be the nth term of the Perrin sequence. The action of the automorphism group of Cn 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…

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## References

SHOWING 1-10 OF 18 REFERENCES

The number of maximal independent sets in connected triangle-free graphs

- MathematicsDiscret. Math.
- 1999

The Fibonacci Number of a Grid Graph and a New Class of Integer Sequences

- Mathematics
- 2005

Given a grid graph G of size mn, we study the number i(m; n) of independent sets in G, as well as b(m; n), the number of maximal such sets. It turns out that the initial cases b(1; n) and b(2; n)…

Maximal independent sets in graphs with at most r cycles

- MathematicsJ. Graph Theory
- 2006

We find the maximum number of maximal independent sets in two families of graphs. The first family consists of all graphs with n vertices and at most r cycles. The second family is all graphs of the…

The number of maximal independent sets in connected graphs

- MathematicsJ. Graph Theory
- 1987

A theorem of Moon and Moser is generalized to determine the maximum number of maximal independent sets in a connected graph on n vertices for n sufficiently large, e.g., n > 50.

Algebraic Graph Theory

- MathematicsGraduate texts in mathematics
- 2001

The Laplacian of a Graph and Cuts and Flows are compared to the Rank Polynomial.

The On-Line Encyclopedia of Integer Sequences

- Computer ScienceElectron. J. Comb.
- 1994

The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences which serves as a dictionary, to tell the user what is known about a particular sequence and is widely used.

Independent sets on path-schemes

- Mathematics
- 2004

We give the generating function for the number of independent sets on the class of well-based path-schemes (a kind of regularly structured graph), which generalizes the known result in this direction.

Alice through Looking Glass after Looking Glass: The Mathematics of Mirrors and Kaleidoscopes

- ArtAm. Math. Mon.
- 2004

Coxeter’s results are described, emphasizing the connection with kaleidoscopes, and some linear algebra (including determinants), basic group theory, and a bit of graph theory are involved.

Mathematics Subject Classification: Primary 05C69; Secondary 05C38

- Mathematics Subject Classification: Primary 05C69; Secondary 05C38
- 1155