ELA, Volume 10, pp. 179-189, July 2003, abstract.
The maximal spectral radius of a digraph with (m+1)^2-s edges
Jan Snellman
It is known that the spectral radius of a digraph with k edges
is at most the square root of k, and that this inequality is strict
except when k is a perfect square. For k=m^2+l, l 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 l/2 (or
greatest integer less than l/2) vertices by pairs of directed edges
(for even l an extra edge is added to the new vertex).
Using a combinatorial reciprocity theorem, and a classification by
Backelin on the digraphs on s edges having a maximal number of walks
of length two, the following result is obtained: for fixed positive s
(s not equal to 4), k=(m+1)^2-s, m large, the maximal spectral radius
of a digraph with k edges is obtained by the digraph which is constructed
from the complete digraph on m+1 vertices by removing the loop at the
last vertex together with s/2 (or greatest integer less than s/2) pairs
of directed edges that connect to the last vertex (if s is even, remove
an extra edge connecting the last vertex).