  There is a deep and interesting connection between the topological properties
of a graph and the behaviour of the dynamical system defined on it. We analyse
various kind of graphs, with different contrasting connectivity or degree
characteristics, using the signless Laplacian matrix. We expose the theoretical
results about the eigenvalue of the matrix and how they are related to the
dynamical system. Then, we perform numerical computations on real-like graphs
and observe the resulting system. Comparing the theoretical and numerical
results we found a perfect consistency. Furthermore, we define a metric which
takes in account the "rigidity" of the graph and enables us to relate all
together the topological properties of the graph, the signless Laplacian matrix
and the dynamical system.

Combinatorics;Computational Physics