# On the reduced Euler characteristic of independence complexes of circulant graphs

## TL;DR

This paper investigates the reduced Euler characteristic of independence complexes of specific circulant graphs, proving non-vanishing in certain cases and providing a counterexample to a conjecture.

## Contribution

It establishes non-zero reduced Euler characteristic for circulant graphs with n=p^k and n=2p^k, and presents a counterexample to a previous conjecture.

## Key findings

- Proved non-vanishing of e for n=p^k and n=2p^k.
- Provided a circulant graph with e=0, countering prior assumptions.
- Enhanced understanding of topological invariants of independence complexes.

## Abstract

Let $G$ be the circulant graph $C_n(S)$ with $S\subseteq\{ 1,\ldots,\left \lfloor\frac{n}{2}\right \rfloor\}$. 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.