Optimal partitions for Robin Laplacian eigenvalues

We prove that the optimal cluster problem for the sum/the max of the first Robin eigenvalue of the Laplacian, in the limit of a large number of convex cells, is asymptotically solved by (the Cheeger sets of) the honeycomb of regular hexagons. The same result is established for the Robin torsional rigidity. In the specific case of the max of the first Robin eigenvalue, we are able to remove the convexity assumption on the cells.

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Geometric Control of the Robin Laplacian Eigenvalues: The Case of Negative Boundary Parameter

Journal of Geometric Analysis ◽

10.1007/s12220-019-00245-9 ◽

2019 ◽

Vol 30 (4) ◽

pp. 4356-4385 ◽

Cited By ~ 1

Author(s):

Dorin Bucur ◽

Simone Cito

Keyword(s):

Geometric Control ◽

Laplacian Eigenvalues ◽

Boundary Parameter ◽

Robin Laplacian

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Correction to: Bounds on Signless Laplacian Eigenvalues of Hamiltonian Graphs

Bulletin of the Brazilian Mathematical Society New Series ◽

10.1007/s00574-020-00240-7 ◽

2021 ◽

Author(s):

Milica Anđelić ◽

Tamara Koledin ◽

Zoran Stanić

Keyword(s):

Hamiltonian Graphs ◽

Laplacian Eigenvalues ◽

Signless Laplacian

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On the Laplacian Eigenvalues of Gn,p

Combinatorics Probability Computing ◽

10.1017/s0963548307008693 ◽

2007 ◽

Vol 16 (6) ◽

pp. 923-946 ◽

Cited By ~ 11

Author(s):

AMIN COJA-OGHLAN

Keyword(s):

Random Graphs ◽

Spectral Gap ◽

Degree Sequences ◽

Laplacian Eigenvalues ◽

Combinatorial Laplacian ◽

Laplacian Spectra

We investigate the Laplacian eigenvalues of sparse random graphs Gnp. We show that in the case that the expected degree d = (n-1)p is bounded, the spectral gap of the normalized Laplacian $\LL(\gnp)$ is o(1). Nonetheless, w.h.p. G = Gnp has a large subgraph core(G) such that the spectral gap of $\LL(\core(G))$ is as large as 1-O (d−1/2). We derive similar results regarding the spectrum of the combinatorial Laplacian L(Gnp). The present paper complements the work of Chung, Lu and Vu [8] on the Laplacian spectra of random graphs with given expected degree sequences. Applied to Gnp, their results imply that in the ‘dense’ case d ≥ ln2n the spectral gap of $\LL(\gnp)$ is 1-O (d−1/2) w.h.p.

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