A Linear Cheeger Inequality using Eigenvector Norms

The Cheeger constant, $h_G$, is a measure of expansion within a graph. The classical Cheeger Inequality states: $\lambda_{1}/2 \le h_G \le \sqrt{2 \lambda_{1}}$where$\lambda_1$ is the first nontrivial eigenvalue of the normalized Laplacian matrix. Hence, $h_G$ is tightly controlled by ${\lambda }_1$ to within a quadratic factor. We give an alternative Cheeger Inequality where we consider the $\mathrm{\infty }$-norm of the corresponding eigenvector in addition to ${\lambda }_1$. This inequality controls $h_G$ to within a linear factor of ${\lambda }_1$ thereby providing an improvement to the previous quadratic bounds. An additional advantage of our result is that while the original Cheeger constant makes it clear that $h_G \to 0$as$\lambda_1 \to 0$,ourresultshowsthat$h_G \to 1/2$as$\lambda_1 \to 1$.