Let G be a connected non-transmission-regular graph with n vertices. Its distance spectral radius lambda_1(G) is the largest eigenvalue of its distance matrix, and D_max(G) is the maximum transmission among its vertices. Liu–Shu–Xue's conjecture. D_max(G)-lambda_1(G)>frac1n+1. This conjecture gives a strict lower bound for the gap between maximum transmission and distance spectral radius in connected non-transmission-regular graphs. The supplied paper is a proof of the conjecture, so the claim is solved.

Liu–Shu–Xue conjecture on distance spectral radius and maximum transmission

The conjecture of Liu, Shu, and Xue asserts a strict lower bound for the gap between maximum vertex transmission and distance spectral radius in every connected non-transmission-regular graph.

Known results

Liu, Shu, and Xue had previously proved the conjecture for trees.

August 29, 2020 proof and 2024 generalization

On August 29, 2020, a preprint claimed a stronger general proof: for odd $n$ , the gap is at least $\frac{n+1-\sqrt{(n-1)(n+3)}}{2}$ , and for even $n$ , at least $\frac{n+2-\sqrt{n^2+4n-4}}{2}$ , with stated extremal graphs. A 2024 paper extends this to $D_\alpha(G)=\alpha Tr(G)+(1-\alpha)D(G)$ and treats the original conjecture as settled. The proof claim remains unverified in this report.

Current status (as of September 2026): The inequality is claimed proved in general, with a later generalization, but independent verification of the proof is not recorded.

Liu–Shu–Xue conjecture on distance spectral radius and maximum transmission

Primary source

Progress summary

Solutions 0