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Abstract
The concept of kpathLaplacian matrix of a graph is motivated and introduced. The pathLaplacian matrices are a natural generalization of the combinatorial Laplacian of a graph. They are defined by using path matrices accounting for the existence of shortest paths of length k between two nodes. This new concept is motivated by the problem of determining whether every node of a graph can be visited by means of a process consisting of hopping from one node to another separated at distance k from it. The problem is solved by using the multiplicity of the trivial eigenvalue of the corresponding kpathLaplacian matrix. We apply these matrices to the analysis of the consensus among agents in a networked system. We show how the consensus in different types of network topologies is accelerated by considering not only nearest neighbors but also the influence of longrange interacting ones. Further applications of pathLaplacian matrices in a variety of other fields, e.g., synchronization, flocking, Markov chains, etc., will open a new avenue in algebraic graph theory.
Original language  English 

Pages (fromto)  33733391 
Number of pages  19 
Journal  Linear Algebra and its Applications 
Volume  436 
Issue number  9 
DOIs  
Publication status  Published  1 May 2012 
Keywords
 graph theory
 path matrices
 laplacian matrix
 consensusanalysis
 synchronization
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Projects
 1 Finished

Mathematics of Large Technological Evolving Networks (MOLTEN)
Higham, D. & Estrada, E.
EPSRC (Engineering and Physical Sciences Research Council)
24/01/11 → 31/03/13
Project: Research