The first two largest eigenvalues of Laplacian, spectral gap problem and Cheeger constant of graphs

We show how the first two largest eigenvalues of Laplacian of a graph or smooth surface can be used to estimate the Cheeger constant of the graph. Particularly, we consider the problem of separating the graph into two large components of approximately equal volumes by making a small cut. This is the idea of Cheeger constant of a graph, which we want to relate to the spectral gap (the difference between the moduli of the first two largest eigenvalues of a Laplacian). We shall use Rayleigh variational characterization of the eigenvalues of the Laplacian to obtain the first two largest eigenvalues. The study reveals that spectral gap of a graph Γ correlates with the Cheeger constant h_Γ of the graph. All our results are illustrated by some simple examples to give a clear insight of the concepts.