On the reduced Euler characteristic of independence complexes of circulant graphs

Abstract

Let $G$ be the circulant graph $C_n(S)$ with $S\subseteq \{1,\dots ,\lfloor \frac{n}{2}\rfloor \}$. We study the reduced Euler characteristic $\tilde{\chi }$ of the independence complex $\Delta (G)$ for $n=p^k$ with $p$ prime and for $n=2p^k$ with $p$ odd prime, proving that in both cases $\tilde{\chi }$ does not vanish. We also give an example of circulant graph whose independence complex has $\tilde{\chi }$ equals to $0$, giving a negative answer to R. Hoshino.