Claims

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1984_12_01_gyarfas_komlos_szemeredi: Gyárfás, Komlós and Szemerédi prove that the reciprocals of the distinct cycle lengths of a graph of minimum degree d sum to at least c log d, which answers the first question; accepted on the refereed J. Graph Theory paper.

2020_10_29_liu_montgomery: Liu and Montgomery's Corollary 1.2 gives every graph of average degree d a harmonic sum of distinct cycle lengths at least (1/2 - o(1)) log d, the sharp constant for the first question; accepted on the refereed JAMS paper.

2026_09_22_milojevic_montgomery_pokrovskiy_sudakov: Milojević, Montgomery, Pokrovskiy and Sudakov (arXiv 2026) prove that for large k the graph K_{k,n-k} uniquely minimizes the harmonic sum of cycle lengths among n-vertex graphs with at least k(n-k) edges.

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