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A note on the stability number of an orthogonality graph. (English) Zbl 1125.05053
Let $$\Omega(n)$$ be the graph on $$2^n$$ vertices corresponding to the vectors $$\{0,1\}^n$$, such that two vertices are adjacent if and only if the Hamming distance between them is $$n/2$$. Then $$\Omega(n)$$ is regular of degree $${n\choose n/2}$$.
Let $$\alpha(\Gamma)$$ be the stability number of a graph $$\Gamma$$. This note studies upper bounds on the stability number $$\alpha(\Omega(n))$$ for $$n\leq 32$$. It is known that $$\alpha(\Omega(n))\leq 2^n/n$$ (the ratio bound). Schrijver (Laurent) obtained new upper bounds $$\overline \alpha(n)$$ ($$l^+(n)$$) such that $$\alpha(\Omega(n))\leq l^+(n)\leq \overline\alpha(n)\leq 2^n/n$$.
For $$n=16$$ the lower and Schrijver bound coincide, so $$\alpha(\Omega(16))=2304$$ (previously known was $$\alpha(\Omega(16))\leq 3912$$). For $$n=20$$ the Schrijver and the Laurent bound coincide, so $$20144\leq \alpha(\Omega(20))\leq 20164$$. For $$n=24$$ the Schrijver bound is 183373 and the Laurent bound is 184194, so $$178208\leq \alpha(\Omega(24))\leq 183173$$.

##### MSC:
 05C35 Extremal problems in graph theory 90C22 Semidefinite programming
##### Keywords:
orthogonality graph; Delsarte bound; stability number
SeDuMi
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##### References:
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