We can similarly define kmin . It is easily seen that the problem of finding kmin
is equivalent to finding a shortest cycle in a digraph and is therefore polynomially
solvable [19]. In Example 1, we have kmin = 2 and kmax = 3.
As usual a real sequence g0, g1 , . . . is called convex [concave] if
gr-1 + gr+1 ≥ 2gr
[gr-1 + gr+1 ≤ 2gr]
for all r = 1, 2, ....
One class of solvable cases of the JRP is related to the fact that δk (A) for
k = 1, 2, . . . , n are the coefficients of the characteristic polynomial of A in max-
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