Bonferroni Inequalities and Negative Cycles in Large Complete Signed Graphs

In this paper the problem of characterizing extremal graphsKnrelatively to the number of negativep-cycles, when the number of negative edges is fixed, is solved for largen.This number can be expressed as an alternating sum for which the Bonferroni inequalities hold. y, the asymptotic value of the probability that ap-cycle ofKnis negative is found asn→∞, if the negative edges induce a subgraph the components of which are paths or cycles.