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  (see also ) 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 , and a general refer-
ence for symmetric matrices is , but there are many others.
 R.B. Bapat and T.E.S. Raghavan, Nonnegative matrices and applications, Encyclope-
dia of Mathematics and its Applications, 64, Cambridge University Press, Cambridge,
 R.A. Brualdi, Spectra of digraphs, Linear Algebra Appl., 432 (2010), 2181–2213.
 R.A. Brualdi and A.J. Hoffman, On the spectral radius of (0,1)-matrices, Linear Alg.
Appl., 65 (1985), 133–146.
 D. Cvetkovi´ c, P. Rowlinson, and S. Simi´ c, Eigenspaces of graphs, Encyclopedia of
Mathematics and its Applications, 66, Cambridge University Press, Cambridge, UK,
 S. Friedland, The maximum eigenvalue of 0-1 matrices with prescribed number of 1s,
Linear Algebra Appl., 69 (1985), 33-69.
 B.N. Parlett, The symmetric eigenvalue problem, Classics in Applied Mathematics,
20, SIAM, Philadelphia, PA, 1998.
 B. Schwarz, Rearrangements of square matrices with non-negative elements, Duke
Math. J., 31 (1964), 45–62.
 P. Rowlinson, On the maximum index of graphs with a prescribed number of edges,
Linear Algebra Appl., 110 (1988), 43–53.