Bonferroni Inequalities and Negative Cycles in Large Complete Signed Graphs

Abstract In this paper the problem of characterizing extremal graphs K n relatively to the number of negative p- cycles, when the number of negative edges is fixed, is solved for large n. This number can be expressed as an alternating sum for which the Bonferroni inequalities hold. Finally, the asymptotic value of the probability that a p- cycle of K n is negative is found as n →∞, if the negative edges induce a subgraph the components of which are paths or cycles.