Line-Graph Signature Beyond the 2-Core: Exact Counterexamples, Rooted Response, and Extremal Constructions at Fixed Cyclomatic Number

This preprint studies the signature of the adjacency matrix of line graphs using the shifted signless Laplacian Q(G) − 2I, rooted-response methods, and reductions to the 2-core. The manuscript proves an exact pendant-forest reduction, a parity property for rooted-tree responses, a singular attachment lemma in the graph-port setting, and a rank-one criterion determining when a pendant leaf increases the line-graph signature. It provides exact counterexamples to a natural monotone 2-core reduction and establishes the lower bound f(c) ≥ floor((c + 1) / 2) for every cyclomatic number c ≥ 1. The corresponding upper bound 2s(L(G)) ≤ c(G) + 1 is stated as a conjecture and remains open. The load-bearing mathematical certificates and the principal computational results were independently checked. Exhaustive exact verification is reproduced through order seven. The order-eight census is retained as an archived campaign result, while the order-nine results are based on numerical screening only. Andrea Paone and Marco Paone contributed equally to this work. Project association: Aletheia Technologies, an independent research project and not an institutional affiliation. Preprint. Not peer reviewed. A related earlier manuscript concerning the original counterexample and the unbounded construction is under editorial consideration separately.