Independent Sets In Association Schemes

Let X be k-regular graph on v vertices and let τ denote the least eigenvalue of its adjacency matrix A(X). If α(X) denotes the maximum size of an independent set in X, we have the following well known bound:

$$ \alpha {\left( X \right)} \leqslant \frac{v} {{1 - \frac{k} {\tau }}} $$

. It is less well known that if equality holds here and S is a maximum independent set in X with characteristic vector x, then the vector

$$ x - \frac{{{\left| S \right|}}} {v}1 $$

is an eigenvector for A(X) with eigenvalue τ . In this paper we show how this can be used to characterise the maximal independent sets in certain classes of graphs. As a corollary we show that a graph defined on the partitions of {1, . . . ,9} with three cells of size three is a core.

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Correspondence to C. D. Godsil*.

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* Researchs upported by NSERC.

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Godsil*, C.D., Newman*, M.W. Independent Sets In Association Schemes. Combinatorica 26, 431–443 (2006).

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Mathematics Subject Classification (2000):

  • 05C69
  • 05E30