Spectrum
The spectrum of the discrete Laplacian is of key interest; since it is a self-adjoint operator, it has a real spectrum. For the convention, the spectrum lies within (as the averaging operator has spectral values in ). The smallest non-zero eigenvalue is denoted and is called the spectral gap. There is also the notion of the spectral radius, commonly taken as the largest eigenvalue.
The eigenvectors don't depend on the convention above (for regular graphs), and are the same as for the averaging operator (as they differ by adding a multiple of the identity), though the eigenvalues differ according to the convention.
For operators that approximate the underlying continuous Laplacian the eigenvalues are a sequence of positive real numbers. The first eigenvalue is zero, if the domain has a boundary and the Neumann boundary condition is used, or if the shape contains no boundary (e.g. the sphere).
Read more about this topic: Discrete Laplace Operator