Geodesic
The maximal spectral radius of a digraph with $(m+1)^2 - s$ edges
The electronic journal of linear algebra, Tome 10 (2003), pp. 179-189.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: It is known that the spectral radius of a digraph with k edges is at most pk, and that this inequality is strict except when k is a perfect square. For k = m2 + `, ` fixed, m large, Friedland showed that the optimal digraph is obtained from the complete digraph on m vertices by adding one extra vertex, a corresponding loop, and then connecting it to the first b`/2c vertices by pairs of directed edges (for even ` an extra edge is added to the new vertex).
Classification : 05C50, 05C20, 05C38
Keywords: spectral radius, digraphs, (0, 1)-matrices, perron-Frobenius theorem, number of walks 
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     author = {Snellman, Jan},
     title = {The maximal spectral radius of a digraph with $(m+1)^2 - s$ edges},
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     year = {2003},
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Snellman, Jan. The maximal spectral radius of a digraph with $(m+1)^2 - s$ edges. The electronic journal of linear algebra, Tome 10 (2003), pp. 179-189. http://geodesic.mathdoc.fr/item/ELA_2003__10__a11/