On the Eigenvalues of Graphs with Mixed Algebraic Structure

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.