Jamison’s questions on the number and average length of subpaths

Jamison’s questions concern extremal counts and average lengths of subpaths in graphs. The retrieved sources connect the minimum-average question to Jamison’s tree result and later conjectures, but do not state every original formulation.

Known results

• Jamison, 1984: the path gives the minimum average connected-subgraph order for trees.
• Kroeker, Mol, and Oellermann, 2018: conjectured the same minimum for all connected graphs.
• Two 2021 preprints proved, for connected graphs of order $n$, $A(G)\geq (n+2)/3$, with equality only for $P_n$.
• A 2025 preprint records the $K_n$ maximum as conjectural and discusses flawed approaches.

August 24, 2026 claimed resolution

A dated report says the preprint The number and average length of subpaths in graphs proves the analogous average-length questions and that $K_n$ maximizes average path length. This is a claimed complete resolution, but the retrieved evidence does not independently verify the proof or reconcile it with the 2025 preprint’s statement that the $K_n$ maximum remained conjectural.

Current status (as of August 2026): The minimum-average result is established, while the claimed proof that $K_n$ maximizes average path length and answers all of Jamison’s questions remains unverified.