2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62606We obtain sharp bounds for the number of n--cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. We prove sharp estimates on both the sum of k-th powers of the coordinates and the Lk norm subject to the constraints that the sum of squares of the coordinates is fixed, and that the sum of the coordinates vanishes.considerable extension of math.CO/9910093CombinatoricsClassical Analysis and ODEs05C38, 49N99text