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arXiv · 2607.22874

The signature of connected line graphs is unbounded

Luke Francis·Trevor Uptain
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Abstract

Akbari, Elphick, Kumar, Pragada and Tang [Discrete Math. 349 (2026) 114953] conjectured that for every connected graph G, the line graph of G has at most one more positive than negative adjacency eigenvalue; equivalently, the signature of a connected line graph is at most 1. We refute the conjecture with two independently found counterexamples: a 14-vertex cactus consisting of two pentagons attached by bridges to adjacent vertices of a square, whose line graph has inertia (9,0,7) by an exact characteristic-polynomial certificate, and a 48-vertex triangle-free graph found by simulated annealing and verified in exact rational arithmetic. Indeed, chaining copies of the 14-vertex graph yields connected graphs on 14k vertices whose line graphs have signature k+1 for every k >= 1. The signature of connected line graphs is therefore unbounded, and no constant-bound repair of the conjecture is possible.

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Luke Francis, Trevor Uptain. 2026-07-24. The signature of connected line graphs is unbounded. https://arxiv.org/abs/2607.22874

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