Characteristic polynomials of production matrices for geometric graphs

Clemens Huemer, Alexander Pilz, Carlos Seara, Rodrigo I. Silveira

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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 languageEnglish
Pages (from-to)631-637
Number of pages7
JournalElectronic Notes in Discrete Mathematics
Volume61
DOIs
Publication statusPublished - 1 Aug 2017
Externally publishedYes

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Keywords

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

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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