### Abstract

An n×n production matrix for a class of geometric graphs has the property that the numbers of these geometric graphs on up to n vertices can be read off from the powers of the matrix. Recently, we obtained such production matrices for non-crossing geometric graphs on point sets in convex position [Huemer, C., A. Pilz, C. Seara, and R.I. Silveira, Production matrices for geometric graphs, Electronic Notes in Discrete Mathematics 54 (2016) 301–306]. In this note, we determine the characteristic polynomials of these matrices. Then, the Cayley-Hamilton theorem implies relations among the numbers of geometric graphs with different numbers of vertices. Further, relations between characteristic polynomials of production matrices for geometric graphs and Fibonacci numbers are revealed.

Original language | English |
---|---|

Pages (from-to) | 631-637 |

Number of pages | 7 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 61 |

DOIs | |

Publication status | Published - 1 Aug 2017 |

Externally published | Yes |

### Fingerprint

### Keywords

- Fibonacci number
- geometric graph
- production matrix
- Riordan array

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Electronic Notes in Discrete Mathematics*,

*61*, 631-637. https://doi.org/10.1016/j.endm.2017.07.017

**Characteristic polynomials of production matrices for geometric graphs.** / Huemer, Clemens; Pilz, Alexander; Seara, Carlos; Silveira, Rodrigo I.

Research output: Contribution to journal › Article › Research › peer-review

*Electronic Notes in Discrete Mathematics*, vol. 61, pp. 631-637. https://doi.org/10.1016/j.endm.2017.07.017

}

TY - JOUR

T1 - Characteristic polynomials of production matrices for geometric graphs

AU - Huemer, Clemens

AU - Pilz, Alexander

AU - Seara, Carlos

AU - Silveira, Rodrigo I.

PY - 2017/8/1

Y1 - 2017/8/1

N2 - An n×n production matrix for a class of geometric graphs has the property that the numbers of these geometric graphs on up to n vertices can be read off from the powers of the matrix. Recently, we obtained such production matrices for non-crossing geometric graphs on point sets in convex position [Huemer, C., A. Pilz, C. Seara, and R.I. Silveira, Production matrices for geometric graphs, Electronic Notes in Discrete Mathematics 54 (2016) 301–306]. In this note, we determine the characteristic polynomials of these matrices. Then, the Cayley-Hamilton theorem implies relations among the numbers of geometric graphs with different numbers of vertices. Further, relations between characteristic polynomials of production matrices for geometric graphs and Fibonacci numbers are revealed.

AB - An n×n production matrix for a class of geometric graphs has the property that the numbers of these geometric graphs on up to n vertices can be read off from the powers of the matrix. Recently, we obtained such production matrices for non-crossing geometric graphs on point sets in convex position [Huemer, C., A. Pilz, C. Seara, and R.I. Silveira, Production matrices for geometric graphs, Electronic Notes in Discrete Mathematics 54 (2016) 301–306]. In this note, we determine the characteristic polynomials of these matrices. Then, the Cayley-Hamilton theorem implies relations among the numbers of geometric graphs with different numbers of vertices. Further, relations between characteristic polynomials of production matrices for geometric graphs and Fibonacci numbers are revealed.

KW - Fibonacci number

KW - geometric graph

KW - production matrix

KW - Riordan array

UR - http://www.scopus.com/inward/record.url?scp=85026744017&partnerID=8YFLogxK

U2 - 10.1016/j.endm.2017.07.017

DO - 10.1016/j.endm.2017.07.017

M3 - Article

VL - 61

SP - 631

EP - 637

JO - Electronic Notes in Discrete Mathematics

JF - Electronic Notes in Discrete Mathematics

SN - 1571-0653

ER -