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 $m_H$ edges and clique number $\omega_H$, $\sum_{i=1}^{\ell}\lambda_i(H)^2\leq 2(1-\omega_H^{-1})m_H$ where $\ell=\min\{n^+(H),\omega_H\}$ and $\lambda_1\geq\cdots$ are the adjacency eigenvalues; the case $\ell=2$ 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 $\operatorname{spec} A(L(G))=\{q_i(G)-2\}_{i}\cup\{-2\}^{m-n+b}$ for the signless Laplacian eigenvalues $q_i(G)$ of $G$ (where $b$ counts bipartite components), we show that the untruncated inequality for $L(G)$ is equivalent to a single inequality $\sum_{q_i(G)\lt 2}\bigl(2-q_i(G)\bigr)^2 \;\geq\;\frac1{\Delta(G)}\sum_{v\in V}f_{\Delta(G)}(d_v), \qquad f_\Delta(d)=d^2-(2\Delta+1)d+4\Delta ,$ 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 $f_\Delta(d)\leq 0$ for every integer $\Delta\geq 3$ and every integer $3\leq d\leq\Delta$, whence the right-hand side is non-positive whenever $\delta(G)\geq 3$. 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 $D\geq3$, 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 $\delta\geq3$ and $n\leq8$ — 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 $\lambda_1(L)^2\leq(1-\omega^{-1})m$, and only when $\Delta\geq9$ or $e\geq56$. 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 ($\Delta\leq8$, $e\leq55$) or prove only a single-eigenvalue bound ($\Delta\geq9$ or $e\geq56$). 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.Version 2 corrects errors found in a full external review: the Cvetkovic spectral identity now states the correct multiplicity m-n+b of the -2 eigenvalue (the earlier form overcounted by b, as K_{1,3} shows); truncated and untruncated square sums are distinguished throughout; cor:F2 is proved structurally for every r-regular line graph with r>=3 (covering non-regular roots such as L(K_{2,4})); the KP comparison is corrected to domination-in-the-tested-range with the correct asymptotics (t ~ n^4/6 for m ~ n^3/2); the novelty claim, the R_split threshold n_0, the subdivision family, and the pendant-edge explanation are all reworded to match what is proved.Version 2 corrects errors found in a full external review: the Cvetkovic spectral identity now states the correct multiplicity m-n+b of the -2 eigenvalue (the earlier form overcounted by b, as K_{1,3} shows); truncated and untruncated square sums are distinguished throughout; cor:F2 is proved structurally for every r-regular line graph with r>=3 (covering non-regular roots such as L(K_{2,4})); the KP comparison is corrected to domination-in-the-tested-range with the correct asymptotics (t ~ n^4/6 for m ~ n^3/2); the novelty claim, the R_split threshold n_0, the subdivision family, and the pendant-edge explanation are all reworded to match what is proved.