Lower bounds for the first eigenvalue of the Steklov problem on graphs

2018/03/23 by Perrin, Hélène · 5 citations
#05C35 #05C50 #15A42 #39A12 #58J50 #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1803.08759

Abstract

We give lower bounds for the first non-zero Steklov eigenvalue on connected graphs. These bounds depend on the extrinsic diameter of the boundary and not on the diameter of the graph. We obtain a lower bound which is sharp when the cardinal of the boundary is 2, and asymptotically sharp as the diameter of the boundary tends to infinity in the other cases. We also investigate the case of weigthed graphs and compare our result to the Cheeger inequality.

Cited by

• Sharp bounds and monotonicity results for Neumann eigenvalues
• The first Steklov eigenvalue bound for graphs of positive genus
• Higher Steklov eigenvalues of graphs on surfaces
• A generalized Cheeger inequality and the Steklov Problem on finite graphs
• Comparison between the first Steklov eigenvalue and algebraic connectivity on trees

Related

• Isoperimetric upper bound for the first eigenvalue of discrete Steklov\n problems
• The first Steklov eigenvalue of planar graphs and beyond
• The first Steklov eigenvalue bound for graphs of positive genus
• Upper bounds for the Steklov eigenvalues on trees
• Upper bounds of Steklov eigenvalues on graphs