Page 138 - Textos de Matemática Vol. 44

128 J. K. MERIKOSKI AND A. KOVACˇEC
Example 5.2. Let
0 1 2
A=1 0 2, λ=3.372.
220
Now u = 3.772 is better than uOS = 3.958 but worse than uBG = 3.464.
Example 5.3. To present a small-dimensional matrix for which u beats uBG, letA=A6 witha=6.Thenu=25.195whileuBG =30.
Example 5.4. Let
3 1 2
A=1 2 2, λ=5.388.
221 Even uF = 6 beats u′ = 6.772.
Example 5.5. Let
1 1 2
A=1 2 2, λ=5.612.
223
Again u′ = 6.772, now better than uOS = 6.954 but worse than uB = 6 and
u′BG = 6.464.
References
[1] A. Brauer, Limits for the characteristic roots of a matrix II, Duke Math. J. 14 (1947), 21–26.
[2] A. Brauer and L. Gentry, Bounds for the greatest characteristic root of an irreducible nonnegative matrix, Linear Algebra Appl. 8 (1974), 105–107.
[3] R. A. Brualdi and A. J. Hoffman, On the spectral radius of (0,1)-matrices, Linear Algebra Appl. 65 (1985), 133–146.
[4] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Pr., 1985.
[5] M. Marcus and H. Minc, A Survey of Matrix Theory and Matrix Inequalities, Dover,
1992. Reprint of edition by Prindle, Weber and Schmidt, 1969.
[6] T. A. Muir, A Treatise on the Theory of Determinants, Revised and enlarged by
W. H. Metzler, Dover, 1960. Reprint of edition by Longmans, Green and Co., 1933.
[7] A. Ostrowski and H. Schneider, Bounds for the maximal characteristic root of a non-
negative irreducible matrix, Duke Math. J. 27 (1960), 547–553.
[8] R. P. Stanley, A bound on the spectral radius of graphs with e edges, Linear Algebra
Appl. 87 (1987), 267–269.
[9] H. Yuan, A bound on the spectral radius of graphs, Linear Algebra Appl. 108 (1988),
135–139.