Friedland, Shmuel Lower bounds for the first eigenvalue of certain M-matrices associated with graphs. (English) Zbl 0784.15010 Linear Algebra Appl. 172, 71-84 (1992). The author gives Cheeger type lower bounds [cf. J. Cheeger [Probl. Anal. Symp. in Honor of Salomon Bochner, Princeton Univ. 1969, 195-199 (1970; Zbl 0212.449)] for the smallest eigenvalue of the Laplacian of corresponding undirected connected multigraphs in terms of the expansion coefficients and norm estimates. Upper bounds for the convergence rate of certain nonnegative irreducible symmetric matrices and irreducible diagonally symmetrizable stochastic matrices are also obtained. Reviewer: P.M.Gibson (Huntsville) Cited in 1 ReviewCited in 9 Documents MSC: 15A42 Inequalities involving eigenvalues and eigenvectors 65F10 Iterative numerical methods for linear systems 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) Keywords:\(M\)-matrices; lower bounds; smallest eigenvalue; Laplacian; undirected connected multigraphs; convergence rate; nonnegative irreducible symmetric matrices; irreducible diagonally symmetrizable stochastic matrices Citations:Zbl 0212.449 PDFBibTeX XMLCite \textit{S. Friedland}, Linear Algebra Appl. 172, 71--84 (1992; Zbl 0784.15010) Full Text: DOI References: [1] Alon, N., eigenvalues and expanders, (Combinatorica, 6 (1986)), 83-96 · Zbl 0661.05053 [2] Cheeger, J., A lower bound for the smallest eigenvalue of the Laplacian, (Gunning, R. C., Problems in Analysis (1970), Princeton U.P), 195-199 · Zbl 0212.44903 [3] Dodziuk, J., Difference equation, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc., 284, 787-794 (1984) · Zbl 0512.39001 [4] S. Friedland, Estimates of the spectral radius of graphs, Linear Multilin. Algebra; S. Friedland, Estimates of the spectral radius of graphs, Linear Multilin. Algebra [5] Golub, G. H.; Van Loan, C. F., Matrix Computations (1989), Johns Hopkins U.P · Zbl 0733.65016 [6] A. Lubotzky, Discrete groups, expanding graphs and invariant measures, to appear.; A. Lubotzky, Discrete groups, expanding graphs and invariant measures, to appear. · Zbl 1183.22001 [7] Sinclair, A.; Jerrum, M., Approximate counting, uniform generation and rapidly mixing Markov chains (1987), Preprint This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.