What was already known
Kathleen Collins conjectured that, for any tree, the nonzero adjacency-polynomial coefficient sequence is unimodal and peaks at the same normalized location as the distance-polynomial coefficient sequence (1989, as quoted in Wagner, Conjecture 2.4). Wagner's paper describes earlier algorithmic searches for extremal combinatorial structures, including his 2020 linear-programming method; this specific 48-vertex tree is presented as a new counterexample.
AI involvement
Wagner selected the conjecture and reward function and designed the deep cross-entropy search. The network generated the retained 48-vertex tree; Wagner then interpreted its structure and proved a broader infinite-family result by hand.
Wagner designed the deep cross-entropy search, selected the reward and interpreted the result; the network produced the retained 48-vertex tree.
Importance
This gives a concrete counterexample to the peak-alignment part of a tree-polynomial conjecture. Wagner used its structure to derive a human-proved infinite family, which is distinct from the network's finite output.
Limitations
The curated AI contribution is the 48-vertex finite tree in Figure 7, not Wagner's later theorem on an infinite family. The paper reports the coefficient-peak calculations, but no separately documented rigorous check of the exact finite instance was located.
Technical verification
The paper reports computed coefficient-peak positions 1/7 and 1/2 for the finite tree but gives no separately documented exact certificate or completed independent check. This is a source-reported finite calculation.
Human understanding
Wagner identifies the tree’s salient high-degree-vertex structure and uses it to derive and prove a broader family, showing human understanding of the finite counterexample’s mechanism.
Correction and publication history
Primary-paper review of the network-produced finite counterexample; local inclusion without notification.
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