Remarks on two conjectures of Li and Guo on Laplacian eigenvalues and degree sequences

Two statements in W.-J. Li and J.-M. Guo, "A new conjecture on Laplacian eigenvalues and degree sequence of a graph", Filomat 39:13 (2025) 4551-4564 (doi:10.2298/FIL2513551L), fail as printed. (a) Conjecture 1.2, Remark 1 inequality (3) and Corollary 2.19 fail for the 5-cycle C5 (at k = 2, 5 + sqrt5 > 4 + sqrt10); this is an exception dropped when 1.2 is derived from 1.1 by the Cauchy-Schwarz inequality, since the paper itself shows that C₄ₖ₊₁ violates 1.1. (b) Conjecture 1.1, stated for every graph other than C₄ₖ₊₁, fails for the disconnected graphs C5 + K1, C5 + K2 and 2C5; the proof of Lemma 2.1, which reduces the conjecture to connected graphs, assumes that every component satisfies the inequality. The note quotes the statements with page numbers, gives exact Laplacian characteristic polynomials and exact decisions in Q(sqrt5), states and proves a corrected Lemma 2.1, and records that the connected case of Conjecture 1.1 remains open. A standard-library Python script verifies every computation exactly. Claude (Anthropic) was used, under the direction of Kerem Özdemir, to search for the examples, to develop the arguments, to write the verification script and to draft this note; the author takes full responsibility for the content.