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The average size of independent sets of graphs

  • Eric O. D. AndriantianaEmail author
  • Valisoa Razanajatovo Misanantenaina
  • Stephan Wagner
Research Article
  • 8 Downloads

Abstract

We study the average size of independent (vertex) sets of a graph. This invariant can be regarded as the logarithmic derivative of the independence polynomial evaluated at 1. We are specifically concerned with extremal questions. The maximum and minimum for general graphs are attained by the empty and complete graph respectively, while for trees we prove that the path minimises the average size of independent sets and the star maximises it. Although removing a vertex does not always decrease the average size of independent sets, we prove that there always exists a vertex for which this is the case.

Keywords

Independent sets Average size Trees Extremal problems 

Mathematics Subject Classification

05C35 05C05 05C07 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics (Pure and Applied)Rhodes UniversityGrahamstownSouth Africa
  2. 2.Department of Mathematical SciencesStellenbosch UniversityMatielandSouth Africa

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