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Echange total, diffusion et quelques résultats sur les itérés de line-digraphs

Abstract : We deal with global communication on connected graphs. First, we consider the case of the total exchange. The minimum total exchange time (M.T.E.T.) is the minimum number of steps required to fully inform all the vertices. We establish new bounds (lower and upper) on the M.T.E.T. We determine the M.T.E.T. of the trees and more generally of the graphs whose minimum degree is one. We prove a conjecture of Bermond, Kodate, Marlin, and Perennes on the fixed points of a complete rotation of some toroidal meshes, and deduce the optimality of their M.T.E.T. We prove also that the M.T.E.T. of a directed De Bruijn graph is optimal. In a second part, we consider the classical broadcast problem. We determine a new upper bound on the broadcast time of an undirected graph, depending of its connectivity. We caractherize some graphs having the same matching number and minimum degree and give their broadcast time. We give new results on the broadcast in De Bruijn graphs. In the last part, we give original results on the independence number of a de Bruijn graph and more generally on the independence number of iterated line digraphs. Then, we prove that the minimum size of a quasi center of a De Bruijn graph UB(d, D) is d - 1, which validates a first conjecture of Bond. At last, we prove that the radius of a Kautz graph UK(2, D) is D, which validates a second conjecture of Bond.
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Submitted on : Wednesday, January 23, 2008 - 5:08:25 PM
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Nicolas Lichiardopol. Echange total, diffusion et quelques résultats sur les itérés de line-digraphs. Interface homme-machine [cs.HC]. Université Nice Sophia Antipolis, 2003. Français. ⟨tel-00214261⟩

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