### Abstract

An independent vertex set of a graph is a set of vertices of the graph in which no two vertices are adjacent, and a maximal independent set is one that is not a proper subset of any other independent set. In this paper we count the number of maximal independent sets of vertices on a complete rectangular grid graph. More precisely, we provide a recursive matrix-relation producing the partition function with respect to the number of vertices. The asymptotic behavior of the maximal hard square entropy constant is also provided. We adapt the state matrix recursion algorithm, recently invented by the author to answer various two-dimensional regular lattice model problems in enumerative combinatorics and statistical mechanics.

Original language | English |
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Pages (from-to) | 2762-2768 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 340 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2017 Dec 1 |

### Keywords

- Enumeration
- Grid graph
- Maximal independent set

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics*,

*340*(12), 2762-2768. https://doi.org/10.1016/j.disc.2017.08.015