Unbounded Signature of Line Graphs: Counterexamples and Transfer Principles

Akbari, Elphick, Kumar, Pragada, and Tang conjectured that every connected graph satisfies a one-unit upper bound between the positive and negative adjacency inertia indices of its line graph. Version 1 exhibited a connected simple counterexample with line-graph inertia (9, 0, 7). Version 2 proves that the failure is unbounded even for connected simple planar subcubic cactus graphs. It establishes a rooted-module attachment lemma and an explicit rooted C4-C5 signature amplifier, yielding a family whose line-graph signature grows by one at each attachment. It also proves an arbitrary-edge integral unimodular four-subdivision congruence preserving determinant, adjacency cokernel, nonunit Smith factors, and nullity over every field. Finally, two independent exact methods classify all 256 residue classes of three-cycle chains and agree row by row.