Geodesic
Inequalities for the extreme eigenvalues of block-partitioned Hermitian matrices with applications to spectral graph theory
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIII, Tome 382 (2010), pp. 82-103 
L. Yu. Kolotilina. Inequalities for the extreme eigenvalues of block-partitioned Hermitian matrices with applications to spectral graph theory. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIII, Tome 382 (2010), pp. 82-103. http://geodesic.mathdoc.fr/item/ZNSL_2010_382_a7/
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     author = {L. Yu. Kolotilina},
     title = {Inequalities for the extreme eigenvalues of block-partitioned {Hermitian} matrices with applications to spectral graph theory},
     journal = {Zapiski Nauchnykh Seminarov POMI},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2010_382_a7/}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $A=D_A+B$ be a block $r\times r$, $r\ge2$, Hermitian matrix of order $n$, where $D_A$ is the block diagonal part of $A$. The main results of the paper are Theorems 2.1 and 2.2, which state the sharp inequalities $$ \lambda_1(A)\ge\lambda_1(D_A+\xi B)\quad\text{and}\quad\lambda_n(A)\le\lambda_n(D_A+\xi B),\qquad-\frac1{r-1}\le\xi\le1, $$ and analyze the equality cases. Some implications of these results are indicated. As applications, matrices occurring in spectral graph theory are considered, and new lower bounds on the chromatic number of a graph are obtained. Bibl. 7 titles.

[1] L. Yu. Kolotilina, “O krainikh sobstvennykh znacheniyakh blochnykh $2\times2$ ermitovykh matrits”, Zap. nauchn. semin. POMI, 296, 2003, 27–38 | MR | Zbl

[2] N. Aronszajn, “Rayleigh-Ritz and A. Weinstein methods for approximation of eigenvalues. I. Operators in a Hilbert space”, Proc. Nat. Acad. Sci. U.S.A., 34 (1948), 474–480 | DOI | MR | Zbl

[3] A. J. Hoffman, “On eigenvalues and colorings of graphs”, Topics in Matrix Analysis, Academic Press, New York, 1970, 79–91 | MR

[4] L. Yu. Kolotilina, “Eigenvalue bounds and inequalities using vector aggregation of matrices”, Linear Algebra Appl., 271 (1998), 139–167 | DOI | MR | Zbl

[5] H. Minc, Nonnegative Matrices, John Wiley and Sons, New York etc., 1988 | MR

[6] V. Nikiforov, “Chromatic number and spectral radius”, Linear Algebra Appl., 426 (2007), 810–814 | DOI | MR | Zbl

[7] B. N. Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall, Inc., 1980 | MR | Zbl