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.