Spectral Gap and Dirichlet Ground State

This chapter is entirely devoted to the studies of the lowest non-trivial eigenvalue of operators on graphs. For standard Laplacians on connected graphs the lowest eigenvalue is \( \lambda _1 = 0 \) and we shall be interested in \( \lambda _2\) , which coincides with the spectral gap \( \lambda _2 - \lambda _1\) . Spectral gap For Laplacians with Dirichlet vertices it is already non-trivial to calculate the ground state \( \lambda _1 > 0 \) . To study these quantities similar methods can be used: Eulerian path and symmetrisation techniques, Cheeger’s approach, surgery principles. Most of these methods work for Schrödinger operators but in order to illuminate connections between spectrum and topology/geometry we shall focus on standard and Dirichlet Laplacians. The methods developed will be extended to higher eigenvalues in the following chapter.