On the reduced Euler characteristic of independence complexes of circulant graphs

Let G be the circulant graph Cn(S) with S⊆{1,…,n2}. We study the reduced Euler characteristic χ̃ of the independence complex Δ(G) for n=pk with p prime and for n=2pk with p odd prime, proving that in both cases χ̃ does not vanish. We also give an example of circulant graph whose independence complex has χ̃ which equals 0, giving a negative answer to R. Hoshino.