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)
only contains 0s and 1s, say e 1s and
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.
[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,
[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