Graphs and Combinatorics

, Volume 33, Issue 5, pp 1103–1118

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

Original Paper

## 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

05C69 05C35

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