Isoperimetric numbers of graphs

Mohar, Bojan

doi:10.1016/0095-8956(89)90029-4

Cited by 294 publications

(207 citation statements)

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“…The computation of i(G) is an NP-hard problem [26]. However, an important result in graph theory gives a lower bound for the isoperimetric number in terms of the second eigenvalue of the Laplacian, namely, i(G) ≥ 1 2 λ 2 .…”

Section: Structural Limitations To Synchronizationmentioning

confidence: 99%

Network synchronization: Spectral versus statistical properties

Atay

Bıyıkoğlu

Jost

2006

Physica D: Nonlinear Phenomena

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We consider synchronization of weighted networks, possibly with asymmetrical connections. We show that the synchronizability of the networks cannot be directly inferred from their statistical properties. Small local changes in the network structure can sensitively affect the eigenvalues relevant for synchronization, while the gross statistical network properties remain essentially unchanged. Consequently, commonly used statistical properties, including the degree distribution, degree homogeneity, average degree, average distance, degree correlation, and clustering coefficient, can fail to characterize the synchronizability of networks.

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Section: Structural Limitations To Synchronizationmentioning

confidence: 99%

Network synchronization: Spectral versus statistical properties

Atay

Bıyıkoğlu

Jost

2006

Physica D: Nonlinear Phenomena

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“…In particular, since λ 2 ≥ k − √ k 2 − i 2 [8,5] where i is the isoperimetric number of the graph, and using the results in [9], a graph of c k for k between 3 and 15 is shown in Figure 1.…”

Section: Random Couplingmentioning

confidence: 99%

Perturbation of coupling matrices and its effect on the synchronizability in arrays of coupled chaotic systems

2003

Physics Letters A

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In a recent paper, wavelet analysis was used to perturb the coupling matrix in an array of identical chaotic systems in order to improve its synchronization. As the synchronization criterion is determined by the second smallest eigenvalue λ2 of the coupling matrix, the problem is equivalent to studying how λ2 of the coupling matrix changes with perturbation. In the aforementioned paper, a small percentage of the wavelet coefficients are modified. However, this result in a perturbed matrix where every element is modified and nonzero. The purpose of this paper is to present some results on the change of λ2 due to perturbation. In particular, we show that as the number of systems n → ∞, perturbations which only add local coupling will not change λ2. On the other hand, we show that there exists perturbations which affect an arbitrarily small percentage of matrix elements, each of which is changed by an arbitrarily small amount and yet can make λ2 arbitrarily large. These results give conditions on what the perturbation should be in order to improve the synchronizability in an array of coupled chaotic systems. This analysis allows us to prove and explain some of the synchronization phenomena observed in a recently studied network where random coupling are added to a locally connected array. Finally we classify various classes of coupling matrices such as small world networks and scale free networks according to their synchronizability in the limit.

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“…A strong improvement over the Alon's discrete version of the Cheeger's inequality was obtained by Mohar [95] in connection with another problem. The isoperimetric number i(G) of a graph G is equal to…”

Section: Isoperimetric Numbersmentioning

confidence: 99%

“…Discrete versions of the Cheeger's bound were found by Alon [5] (vertex version as mentioned above), Dodziuk [41] (for infinite graphs), Varopoulos [130] (also for the infinite case), Mohar [94,95], Sinclair and Jerrum [124] (in terms of Markov chains), Friedland [49]. Cf.…”

Section: Isoperimetric Numbersmentioning

confidence: 99%