Graphs and Combinatorics

, Volume 33, Issue 5, pp 1103–1118 | Cite as

Maximizing the Number of Independent Sets of Fixed Size in Connected Graphs with Given Independence Number

Original Paper
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Abstract

The Turán connected graph \(\mathrm {TC}_{n,\alpha }\) is obtained from \(\alpha \) cliques of size \(\lfloor \frac{n}{\alpha } \rfloor \) or \(\lceil \frac{n}{\alpha } \rceil \) by joining all cliques by an edge to one central vertex in one of the larger cliques. The graph \(\mathrm {TC}_{n,\alpha }\) was shown recently by Bruyère and Mélot to maximise the number of independent sets among connected graphs of order n and independence number \(\alpha \). We prove a generalisation of this result by showing that \(\mathrm {TC}_{n,\alpha }\) in fact maximises the number of independent sets of any fixed cardinality \(\beta \le \alpha \). Several results (both old and new) on the number of independent sets or maximum independent sets follow as corollaries.

Keywords

Independent sets Independence number Connected graphs Turán connected graphs 

Mathematics Subject Classification

05C69 05C35 

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Copyright information

© Springer Japan KK 2017

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.Department of Mathematical SciencesStellenbosch UniversityMatielandSouth Africa

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