BIBLIOGRAPHY 11 For nonsymmetric matrices the analogue of the equality condition in Theorem 1.9 is not true in general. See Example 1.5. In case c1,c2,...,c(n) 2 only contains 0s and 1s, say e 1s and ( n 2 ) e 0s, then the matrices in Sn are the adjacency matrices of graphs with n vertices and e edges (no loops). Then Theorem 1.9 asserts that in order for a matrix in Sn to achieve the largest λmax, it must have the property that whenever apq = 1 with p q, then aij = 1 for all i and j with i j and i p and j q. For each e, Rowlinson [8] (see also [4]) determined the matrices with the largest λmax, that is, those graphs on n vertices and e edges with the largest λmax of their adjacency matrices. A general reference for nonnegative matrices is [1], and a general refer- ence for symmetric matrices is [6], but there are many others. Bibliography [1] R.B. Bapat and T.E.S. Raghavan, Nonnegative matrices and applications, Encyclope- dia of Mathematics and its Applications, 64, Cambridge University Press, Cambridge, UK, 1997. [2] R.A. Brualdi, Spectra of digraphs, Linear Algebra Appl., 432 (2010), 2181–2213. [3] R.A. Brualdi and A.J. Hoffman, On the spectral radius of (0,1)-matrices, Linear Alg. Appl., 65 (1985), 133–146. [4] D. Cvetkovi´ c, P. Rowlinson, and S. Simi´ c, Eigenspaces of graphs, Encyclopedia of Mathematics and its Applications, 66, Cambridge University Press, Cambridge, UK, 1997. [5] S. Friedland, The maximum eigenvalue of 0-1 matrices with prescribed number of 1s, Linear Algebra Appl., 69 (1985), 33-69. [6] B.N. Parlett, The symmetric eigenvalue problem, Classics in Applied Mathematics, 20, SIAM, Philadelphia, PA, 1998. [7] B. Schwarz, Rearrangements of square matrices with non-negative elements, Duke Math. J., 31 (1964), 45–62. [8] P. Rowlinson, On the maximum index of graphs with a prescribed number of edges, Linear Algebra Appl., 110 (1988), 43–53.
Previous Page Next Page