A Linear Cheeger Inequality using Eigenvector Norms

The Cheeger constant, hG, is a measure of expansion within a graph. The classical Cheeger Inequality states: λ1/2 ≤ hG ≤ √ 2λ1 where λ1 is the first nontrivial eigenvalue of the normalized Laplacian matrix. Hence, hG is tightly controlled by λ1 to within a quadratic factor. We give an alternative Cheeger Inequality where we consider the∞-norm of the corresponding eigenvector in addition to λ1. This inequality controls hG to within a linear factor of λ1 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 hG → 0 as λ1 → 0, our result shows that hG → 1/2 as λ1 → 1.