  Using methods from algebraic graph theory and convex optimization, we study
the relationship between local structural features of a network and spectral
properties of its Laplacian matrix. In particular, we derive expressions for
the so-called spectral moments of the Laplacian matrix of a network in terms of
a collection of local structural measurements. Furthermore, we propose a series
of semidefinite programs to compute bounds on the spectral radius and the
spectral gap of the Laplacian matrix from a truncated sequence of Laplacian
spectral moments. Our analysis shows that the Laplacian spectral moments and
spectral radius are strongly constrained by local structural features of the
network. On the other hand, we illustrate how local structural features are
usually not enough to estimate the Laplacian spectral gap.
