The line graph case of the Elphick--Linz--Wocjan conjecture: an exact reduction and a minimum-degree criterion

Let G be a simple graph and $L(G)$ its line graph. Elphick, Linz and Wocjan conjectured that for every graph H with mH edges and clique number ωH, ∑ᵢ₌₁^λᵢ(H)²≤ 2(1-ωH⁻¹)mH where =min⁺(H),ωH\ and λ₁≥⋯ are the adjacency eigenvalues; the case =2 is the Bollob\'as–Nikiforov conjecture, recently resolved in full by Coutinho, Liu, Spier, Tang and Zhang. We make two contributions to the case $H=L(G)$. First, using the classical spectrum identity spec A(L(G))=ᵢ(G)-2\ᵢ∪\-2\ᵐ⁻ⁿ⁺ᵇ for the signless Laplacian eigenvalues qᵢ(G) of G (where b counts bipartite components), we show that the untruncated inequality for $L(G)$ is equivalent to a single inequality ∑qᵢ(G)< 2(2-qᵢ(G))² \;≥\;1Δ(G)∑v∈ VfΔ(G)(dᵥ), f_Δ(d)=d²-(2Δ+1)d+4Δ , whose right-hand side depends only on the degree sequence. This removes colouring, clique computation, truncation and the padding multiplicity $m-n+b$ from the problem entirely. Second, we prove that f_Δ(d)≤ 0 for every integer Δ≥ 3 and every integer 3≤ d≤Δ, whence the right-hand side is non-positive whenever δ(G)≥ 3. Consequently the Elphick–Linz–Wocjan conjecture holds for $L(G)$ for every graph G with minimum degree at least $3$, with no further hypotheses; in particular it holds for every D-regular G with D≥3, a partial answer to a question the original authors state as open. We must state plainly how much of this is new. Zeng and Zhang already prove the untruncated bound for class-1 line graphs with no minimum-degree hypothesis at all, which is strictly more general than our result on that part of the domain. An exact edge-colouring computation over all $2762$ connected graphs with δ≥3 and n≤8 — our whole domain — shows that $2735$ of them are class 1 and only $27$, i.e. $1.0%$, are class 2. Those $27$ are precisely the graphs on which Zeng–Zhang are silent: for class 2 they prove only the single-eigenvalue bound λ₁(L)²≤(1-ω⁻¹)m, and only when Δ≥9 or e≥56. Our genuinely new content is therefore exactly this: the multi-term untruncated bound for class-2 line graphs of minimum degree at least $3$, on which Zeng–Zhang are either silent (Δ≤8, e≤55) or prove only a single-eigenvalue bound (Δ≥9 or e≥56). It is proved uniformly in n by a one-line sign argument on a single quadratic rather than by case analysis. We claim nothing beyond it. We verify the reduction exhaustively on all $273,171$ in-scope connected graphs with at most $9$ vertices, and on all $11,716,571$ connected graphs with $10$ vertices — the complete population for that order, matching the OEIS count A001349(10) exactly, not a sample — with no failures, and we record six proof routes that we attempted and that provably do not work — in particular the natural pendant-count sharpening, which fails on $1,308$ of $12,094$ graphs, showing that the negative high-degree terms are load-bearing.