Lin et al.'s vertex-count conjecture for ABC-minimal trees

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Let tt be the number of leaves of a tree, and call a tree tt-minimal if it has tt leaves and no other tree with the same number of leaves has a smaller ABC-index, where

ABC(G)=∑uv∈E(G)du+dv−2dudv\mathcal{ABC}(G)=\sum_{uv\in E(G)}\sqrt{\frac{d_u+d_v-2}{d_ud_v}}

for a graph GG with vertex degrees dvd_v. Lin et al.'s conjecture. For t≥88t\geq 88, a tt-minimal tree has t+⌊t11⌋−1t+\left\lfloor\tfrac{t}{11}\right\rfloor-1 vertices. This conjecture predicts the order of every ABC-minimal tree with sufficiently many leaves; the source describes it as arising from computer-aided calculations, while the available context does not establish its resolution.

References

Primary source

Bojan Mohar, “The structure of ABC-minimal trees with given number of leaves”, arXiv:1706.02891 (2018).

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