Counterexamples to a conjecture of Zhou, Du and He on the two largest signless Laplacian eigenvalues

Zhou, Du and He (arXiv:2504.04389; Discrete Appl. Math. 384 (2026) 139-144, Conjecture 1) conjectured that among connected graphs with n vertices and cyclomatic number c (1 <= c <= n-2) the graph G(n-2-c, c) = K_1 v (K1,c u (n-2-c)K_1) is the unique maximiser of q_1 + q_2, the sum of the two largest signless Laplacian eigenvalues. This note shows that the conjecture is false. For n = 5 and c = 2 the bowtie K_1 v 2K_2 has q_1 + q_2 = (13 + sqrt17)/2 = 8.5616, larger than 8.4893 for G(1, 2). For c = 3 and every n >= 5 the graph K_1 v (K_3 u (n-4)K_1) is signless-Laplacian cospectral with G(n-5, 3), so the maximiser is not unique; exhaustive search shows that both graphs attain the maximum for n <= 10, and that these are the only failures among connected graphs on at most 10 vertices. A standard-library Python script checks everything in exact rational arithmetic. Searches and verification were carried out with Claude (Anthropic) under the direction of Kerem Özdemir.