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Cut-off for large sums of graphs. (English) Zbl 1135.05039
Summary: If $$L$$ is the combinatorial Laplacian of a graph, $$\exp(-Lt)$$ converges to a matrix with identical coefficients. The speed of convergence is measured by the maximal entropy distance. When the graph is the sum of a large number of components, a cut-off phenomenon may occur: before some instant the distance to equilibrium tends to infinity; after that instant it tends to 0. A sufficient condition for cut-off is given, and the cut-off instant is expressed as a function of the gap and eigenvectors of components. Examples include sums of cliques, stars and lines.

##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 60J27 Continuous-time Markov processes on discrete state spaces
##### Keywords:
Laplacian; sum of graphs; spectrum; Kullback distance; cut-off
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##### References:
 [1] Aldous, D.; Diaconis, P., Shuffling cards and stopping times, Amer. Math. Monthly, 93, 5, 333-348, (1986) · Zbl 0603.60006 [2] Austin, D.; Gavlas, H.; Witte, D., Hamiltonian paths in Cartesian powers of directed cycles, Graphs and Combinatorics, 19, 4, 459-466, (2003) · Zbl 1032.05069 [3] Barrera, J.; Lachaud, B.; Ycart, B., Cutoff for n-tuples of exponentially converging processes, Stoch. Proc. Appl., 116, 10, 1433-1446, (2006) · Zbl 1103.60023 [4] Bellman, R., Introduction to matrix analysis, (1960), McGraw-Hill, London · Zbl 0124.01001 [5] Bezrukov, S. L.; Elsässer, R., Edge-isoperimetric problems for Cartesian powers of regular graphs, Theor. Comput. Sci., 307, 3, 473-492, (2003) · Zbl 1070.68114 [6] Chung, F.; Oden, K., Weighted graph Laplacians and isoperimetric inequalities, Pacific J. of Math., 192, 257-274, (2000) · Zbl 1009.05095 [7] Çinlar, E., Introduction to stochastic processes, (1975), Prentice Hall, New York · Zbl 0341.60019 [8] Colin de Verdière, Y., Spectres de graphes, 4, (1998), SMF · Zbl 0913.05071 [9] Colin de Verdière, Y.; Pan, Y.; Ycart, B., Singular limits of Schrödinger operators and Markov processes, J. Operator Theory, 41, 151-173, (1999) · Zbl 0990.47013 [10] Cvetković, D.; Doob, M.; Gutman, I.; Torgašev, A., Recent results in the theory of graph spectra, 36, (1988), North-Holland, Amsterdam · Zbl 0634.05054 [11] Cvetković, D.; Doob, M.; Sachs, H., Spectra of graphs - Theory and application, (1980), Academic Press, New York · Zbl 0458.05042 [12] Mohar, B., Laplace eigenvalues of graphs - a survey, Discrete Math., 109, 171-183, (1992) · Zbl 0783.05073 [13] Pollard, D., User’s guide to measure theoretic probability, (2001), Cambridge University Press · Zbl 0992.60001 [14] Saloff-Coste, L.; Kesten, H., Random walks on finite groups, Probability on discrete structures, 110, 263-346, (2004), Springer, Berlin · Zbl 1049.60006 [15] Ycart, B., Cutoff for samples of Markov chains, ESAIM Probab. Stat., 3, 89-107, (1999) · Zbl 0932.60077
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