A Counterexample to a Line-Graph Inertia Conjecture

Akbari, Elphick, Kumar, Pragada and Tang conjectured in Conjecture 4.12 of “A new conjecture on the inertia of graphs” (Discrete Mathematics 349 (2026), Article 114953; arXiv:2508.01163v3) that, for every connected graph G, its line graph L(G) satisfies

n⁺(L(G)) ≤ n⁻(L(G)) + 1,

or equivalently

s(L(G)) ≤ 1.

The original paper notes that connectedness is essential: without it, the disjoint union C₅ ∪ C₅ is already a counterexample.

This preprint presents a connected simple graph G on 14 vertices and 16 edges that violates Conjecture 4.12. The graph consists of three cycles, C₅, C₄ and C₅, joined in a chain by two bridges incident with adjacent vertices of the central C₄.

Its line graph L(G) is connected and has exact adjacency inertia

In(L(G)) = (n⁺, n⁰, n⁻) = (9, 0, 7).

Consequently,

n⁺(L(G)) = 9 > 8 = n⁻(L(G)) + 1,

so G is a counterexample to Conjecture 4.12.

The graph is specified by the graph6 string

Mhe?GCD?_?_@?@?C_

and by the following edge description:

C₅: 0–1–2–3–4–0
C₄: 5–6–7–8–5
C₅: 9–10–11–12–13–9
Bridges: 0–5 and 6–9

Moreover,

det A(L(G)) = −16,

so zero is not an eigenvalue. Over the integers, the characteristic polynomial of A(L(G)) factors as

(x − 2)(x − 1)(x + 2)²
(x² + x − 1)²
(x³ − 2x² − 4x + 1)
(x⁵ − x⁴ − 6x³ + 3x² + 7x − 2).

The result is certified through multiple independent exact-arithmetic routes. These include:

1. direct construction of the line graph, exact computation of its characteristic polynomial and rational Sturm root-sign certificates;

2. an independent unsigned-incidence and signless-Laplacian verification based on

   A(L(G)) = BᵀB − 2I,

   together with exact rational-congruence certificates.

The deposited materials include the candidate graph, the complete adjacency matrix of the line graph, machine-readable exact certificates, deterministic verification code, automated tests and cryptographic checksums.

This result concerns the line-graph conjecture stated as Conjecture 4.12. It is distinct from the counterexamples of Chen and Li (arXiv:2605.07196) to the general graph-inertia conjecture

2n⁺(G) ≤ n⁻(G)(n⁻(G) + 1).

To the best of the author’s knowledge, this is the first publicly reported counterexample to Conjecture 4.12. This record contains a preprint and reproducibility materials and has not undergone formal peer review.