Dynamical systems on graphs through the signless Laplacian matrix
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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 into 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.
KeywordsGraph theory Dynamical systems Signless Laplacian Spectral properties Centrality measures Mathematical models
Mathematics Subject Classification68R10
We would like to express our gratitude to Prof. Paolo Cermelli, who supported and encouraged us along all this work, from when it started as two Master thesis until this last version as a paper. Really thanks for being so helpful and still let us work on our on. He taught us a lot, and there is no greater compliment for a professor.
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