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Let C(n) denote the maximum number of induced copies of 5-cycles in graphs on n vertices. For n large enough, we show thatand a, b, c, d, e are as equal as possible.Moreover, if n is a power of 5, we show that the unique graph on n vertices maximizing the number of induced 5-cycles is an iterated blow-up of a 5-cycle.The proof uses flag algebra computations and stability methods.

An assignment of positive integer weights to the edges of a simple graph G is called irregular, if the weighted degrees of the vertices are all different. The irregularity strength, s(G), is the maximal weight, minimized over all irregular assignments. In this study, we show that s(G) c 1 n /, for graphs with maximum degree Á n 1=2 and minimum ------------------

A weighting of the edges of a graph is called irregular if the weighted degrees of the vertices are all different. In this note we show that such a weighting is possible from the weight set {1, 2, . . . , 6 n δ } for all graphs not containing a component with exactly 2 vertices or two isolated vertices.

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