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 χ˜ of the independence complexΔ(G) for n = pk with p prime and for n = 2pk with p odd prime, proving that in both cases χ˜ does not vanish. We also give an example of circulant graph whose independence complex has χ˜ which equals 0, giving a negative answer to R. Hoshino.