  We study some spectral properties of a matrix that is constructed as a
combination of a Laplacian and an adjacency matrix of simple graphs. The matrix
considered depends on a positive parameter, as such we consider the
implications in different regimes of such a parameter, perturbative and beyond.
Our main goal is to relate spectral properties to the graph's configuration, or
to basic properties of the Laplacian and adjacency matrices. We explain the
connections with dynamic networks and their stability properties, which lead us
to state a conjecture for the signature.
