REVIEW 3 major objections 4 minor 14 references
Rank-Average Degree Bound for Graph Energy
T0 review · 3 major / 4 minor · reviewed 2026-08-27 · deepseek-v4-flash
Pith's one-line read Every finite simple graph of order at least five has energy at least its adjacency rank plus its average degree minus one, with equality only for complete graphs and perfect matchings.
desk verdict Proves a strong new rank-average-degree inequality that settles five conjectures; sound modulo unverified external preprints and one repairable gap in the p=2 case. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing objects are the vertex-energy deletion lemma, which shows that $\mathcal{E}(G)-\bar d(G)$ does not increase when a vertex is removed while the adjacency rank is held fixed; the determinant-integrality principle, which says that the pseudodeterminant of an integral adjacency matrix of rank $r$ is a nonzero integer, hence the product of the nonzero eigenvalues has absolute value at least $1$; and a determinant-variance reduction that turns the assumed failure of the bound into a purely analytic inequality in parameters $k,a,w$. The resulting scalar inequality $F(k,a,w)<0$ is then proved by elementary calculus, with the two-positive-eigenvalue case handled separately through a sparse-complement argument.
What would settle it
A concrete calculation: the path $P_4$ fails the inequality because $\mathcal{E}(P_4)=2\sqrt5\approx 4.472$ is just below $4.5 = r(\bar d)(P_4)-1$, which is why the theorem requires $n\ge 5$; a computer search over all graphs on five and six vertices would verify or refute the claimed universal bound in the smallest cases.
Extended reading notes
Core claim
The central claim is that for every finite simple graph with $n\ge 5$ vertices, the energy $\mathcal{E}(G)=\sum_i |\lambda_i|$ is bounded below by the adjacency rank plus the average degree minus one. This is a unification: for nonsingular graphs, whose adjacency matrix is invertible, the rank equals $n$, giving $\mathcal{E}(G)\ge n-1+\bar d(G)$, and since $n-1+\bar d(G)$ dominates both $\Delta(G)+\delta(G)$ and $2\sqrt{\bar d(G)(n-1)}$, the older conjectures follow. The author further claims that equality occurs only for complete graphs and, in even orders, for perfect matchings, and uses this to prove that certain Zagreb-index inequalities hold with equality only for $K_n$.
Load-bearing premise
The proof leans on three previously established results about graph spectra that this paper does not re-derive; if any of those results is wrong or incomplete, the main inequality is not established.
Editorial extensions
If this is right
- Every nonsingular graph of order at least five satisfies $\mathcal{E}(G)\ge n-1+\bar d(G)$, settling the Akbari-Dabirian-Ghasemi conjecture.
- Every nonsingular graph satisfies $\mathcal{E}(G)\ge \Delta(G)+\delta(G)$ and $\mathcal{E}(G)\ge 2\sqrt{\bar d(G)(n-1)}$, with equality only for $K_n$.
- For nonsingular graphs, $\mathcal{E}(G)\ge M_1(G)/m$ and $\mathcal{E}(G)\ge M_1(G)/(2m)+2m/n$, with equality only for $K_n$.
- Combining Theorem 1.1 with rank bounds yields energy lower bounds in terms of clique number, induced matching number, total domination number, and diameter.
- The equality characterization is sharp: the only graphs attaining equality in the rank-average-degree bound are complete graphs and perfect matchings.
Reading between the lines
- Beyond the paper: if the rank bound is accepted as a master inequality, then any future improvement in lower bounds for the adjacency rank, for example via independence or domination parameters, automatically becomes an energy bound through Theorem 1.1.
- Beyond the paper: the determinant-variance reduction suggests a template for converting a nonsingular energy lower bound whose proof uses $|\det A|\ge 1$ into a rank bound by deletion monotonicity; applying the same template to other spectral invariants is a testable scheme.
- Beyond the paper: the sharp order restriction $n\ge 5$ raises the question of a corrected statement for $n=4$; the paper notes that $P_4$ fails, so enumerating all four-vertex graphs would show whether a simple exception list exists.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves a rank–average-degree energy bound: for every finite simple graph G of order n≥5, E(G) ≥ r(G)+d̄(G)−1, with equality exactly for K_n and, when n is even, (n/2)K_2. The proof proceeds by establishing a vertex-deletion monotonicity for E−d̄, treating the nonsingular case with a determinant–variance reduction and an analytic scalar inequality, handling the p=2 case through the complement, and then reducing singular graphs to induced nonsingular subgraphs. As consequences, the paper derives five previously conjectured lower bounds for nonsingular graph energy, including the Δ+δ and Zagreb-index bounds.
Significance. If sound, this is a substantial result: it unifies several open conjectures under one sharp inequality and gives a complete extremal characterization. The paper is genuinely self-contained in its analytic parts: the scalar inequalities are proved with explicit elementary estimates and no fitted parameters, and the base cases are handled carefully. The main caveat is that the nonsingular proof leans on three recent preprints ([2], [10], [11]) and the equality characterization on the author's own preprint [12]; those dependencies are outside the verification offered here.
major comments (3)
- [Section 6 (Proposition 6.1)] The p(G)=2 case is not rigorously completed. The interlacing argument in the first paragraph is valid only if the induced odd cycle is in G; if, as the surrounding notation suggests, it is in the complement of G, then p(complement)≥3 does not contradict p(G)=2, because interlacing for induced subgraphs of the complement controls the inertia of the complement, not of G. The same problem occurs in the 2P3 paragraph: the displayed characteristic polynomial is that of two disjoint copies of P3 in the complement, whereas interlacing in G would require the positive inertia of the complementary induced subgraph, which is never computed. Consequently, the conclusion that the complement is a forest with at most one component of order at least three is not established, and the subsequent double-star/star analysis rests on an unproved structural assumption. The case may be repairable, but as written it is a load-bearing gap.
- [Sections 2, 4, and 8] Theorem 1.1 depends essentially on three external results that are used as black boxes: inequality (4) from [10], inequality (5) from [11], and the two-range nonsingular bounds from [2], which are used to restrict counterexamples to ρ>7.11 and a<logn+1 and to control the determinant–variance contradiction in Proposition 5.6. Since [2], [10], and [11] are recent preprints and a gap in any one of them would invalidate the central theorem, the paper should either prove these results or explicitly state that the main theorem is conditional on them.
- [Section 8 (Lemma 8.2)] The equality characterization of Theorem 1.2 relies on the author's own preprint [12, Theorem 1.3] for the classification of graphs with E(G)=2(n−α(G)). Because Theorem 1.2 is a central claim and [12] is not independently verified in this manuscript, the equality result should not be presented as unconditional; at minimum the relevant classification theorem should be stated in full or proved.
minor comments (4)
- [Section 6] The overline notation for the complement is missing in several displays (for example, 'B=A( G)' and 'm=|E( G)|'), making it hard to follow which graph is meant; please correct the notation throughout.
- [Section 6] After showing that a cyclic component yields strict inequality, the sentence 'Therefore every component of the complement is acyclic' should be phrased as 'It remains to consider the case where every component is acyclic'; without an explicit counterexample assumption, the strict inequality already finishes that case.
- [Section 1] The phrase 'in their stated ranges' for the five consequences is vague; please specify the exact order restriction n≥5 for each of the five inequalities.
- [Section 8 (Lemma 8.4)] The proof is dense but self-contained; consider adding a short table of the endpoint values for a=1,2,3 to improve readability.
Circularity Check
No circular reduction; the main inequality is proved from independent spectral estimates and external theorems, with a minor self-citation [12] confined to the equality characterization.
full rationale
The derivation does not fit any parameter, rename a known result, or define its conclusion into its assumptions. The central inequality E(G) >= r(G)+dbar(G)-1 is obtained from spectral identities (equation 3), the external bounds (4) and (5), the two-range nonsingular theorems of [2], and new analytic lemmas (vertex-energy estimate, deletion monotonicity, determinant-variance reduction, scalar inequality). The nonsingular-to-singular reduction by rank-preserving deletion is a genuine induction, not an identity. The only self-citation is [12], used in Lemma 8.2 to classify graphs satisfying E(G)=2(n-alpha(G)); this classification is not the target inequality and is invoked only in the equality branch of Theorem 1.2, not in the proof of Theorem 1.1. That the paper depends on unverified external preprints [10], [11], and [2] is a correctness risk, not a circularity reduction. A possible proof gap in Proposition 6.1 is also a correctness issue, not a circularity issue. Accordingly no circular step is exhibited, and the paper receives a low score for a non-load-bearing self-citation.
Assumptions & free parameters
assumptions (7)
- domain assumption E(G) ≥ 2(n - α(G)) for every graph G (Kumar and Pragada, 2026, arXiv:2607.19817)
- domain assumption For connected G, min{s+(G), s-(G)} ≥ n-1 (Liu, Tang, and Zhang, 2026, arXiv:2607.18031)
- domain assumption For nonsingular G of order ≥5, the claimed bound holds when λ1(G) ≤ 7.11 or d̄(G) ≤ n - 2 log n - 3 (Akbari, Dabirian, Ghasemi, arXiv:2207.04599)
- domain assumption Classification of graphs with E(G) = 2(n - α(G)) (Mojallal, arXiv:2608.04367)
- standard math De Caen's inequality: M1(G) ≤ m(n - 2 + 2m/(n-1))
- standard math Total domination bound γt(H) ≤ |V(H)| - m_H(0) for non-complete components (Abiad et al., 2023)
- standard math Standard spectral facts: interlacing, Ky Fan's principle, AM-GM, integrality of pseudodeterminant
Cite this review
Pith. "Pith review of Rank-Average Degree Bound for Graph Energy." pith.science (2026). https://pith.science/paper/QT2OOLLN
@misc{pith2026260822139,
author = {Pith},
title = {Pith review of: Rank-Average Degree Bound for Graph Energy},
year = {2026},
howpublished = {\url{https://pith.science/paper/QT2OOLLN}},
note = {Machine review of arXiv:2608.22139}
}
abstract
We prove that the energy ${\mathcal E}(G)$ of any simple graph $G$ of order $n\ge5$ satisfies \[ {\mathcal E}\ge r(G)+\bar d(G)-1, \] where $r(G)$ and $\bar d(G)$ denote, respectively, the rank of the adjacency matrix and the average degree of $G$. We also characterize all extremal graphs. As consequences, our result settles five previously conjectured lower bounds for the energy of nonsingular graphs in their stated ranges, namely \[ \begin{gathered} {\mathcal E}(G)\ge n-1+\bar d(G),\qquad {\mathcal E}(G)\ge\Delta(G)+\delta(G),\qquad {\mathcal E}(G)\ge2\sqrt{\bar d(G) (n-1)},\qquad {\mathcal E}(G)\ge\frac{M_1(G)}{m},\qquad {\mathcal E}(G)\ge\frac{M_1(G)}{2m}+\frac{2m}{n}, \end{gathered} \] where $m$ is the number of edges, $\Delta(G)$ and $\delta(G)$ are the maximum and minimum degrees, and the first Zagreb index $ M_1(G)$ is the sum of degree squares.
Reference graph
Works this paper leans on
-
[10]
Hitesh Kumar and Shivaramakrishna Pragada,Energy and independence number, 2026, arXiv:2607.19817 [math.CO],https://arxiv.org/abs/2607.19817
work page Pith review arXiv 2026
-
[11]
Yinchen Liu, Quanyu Tang, and Shengtong Zhang,The positive and negative square-energy conjecture, 2026, arXiv:2607.18031 [math.CO],https://arxiv.org/abs/2607.18031
work page Pith review arXiv 2026
-
[2]
A lower bound of the energy of non-singular graphs in terms of average degree
Saieed Akbari, Hossein Dabirian, and S. Mahmood Ghasemi,A lower bound of the energy of non-singular graphs in terms of average degree, 2022, arXiv:2207.04599 [math.CO],https: //arxiv.org/abs/2207.04599
work page Pith review arXiv 2022
-
[12]
Seyed Ahmad Mojallal,Extremal graphs for the energy–independence number inequality, 2026, arXiv:2608.04367 [math.CO],https://arxiv.org/abs/2608.04367
work page Pith review arXiv 2026
-
[1]
Aida Abiad, Saieed Akbari, Mohammad Hossein Fakharan, and Ahmad Mehdizadeh,A bound for thep-domination number of a graph in terms of its eigenvalue multiplicities, Linear Algebra Appl.658(2023), 319–330
work page 2023
-
[3]
Saieed Akbari and Mohammad Ali Hosseinzadeh,A short proof for graph energy is at least twice of minimum degree, MATCH Commun. Math. Comput. Chem.83(2020), 631–633
work page 2020
-
[4]
Rajendra Bhatia,Matrix analysis, Graduate Texts in Mathematics, vol. 169, Springer, New York, 1997
work page 1997
-
[5]
Andries E. Brouwer and Willem H. Haemers,Spectra of graphs, Universitext, Springer, New York, 2012
work page 2012
Show all 14 references
-
[6]
Kinkar Chandra Das and Ali Ghalavand,On the connection between energy and zagreb indices of graphs, J. Appl. Math. Comput.71(2025), no. 3, 3555–3575
2025
-
[7]
Dominique de Caen,An upper bound on the sum of squares of degrees in a graph, Discrete Math.185(1998), 245–248
1998
-
[8]
Math.-Statist
Ivan Gutman,The energy of a graph, Ber. Math.-Statist. Sekt. Forschungszentrum Graz103 (1978), 1–22
1978
-
[9]
Akbar Jahanbani and Ivan Gutman,Exact variance-energy relations and optimal spectral bounds for graphs, Open J. Math. Sci.9(2025), 301–307
2025
-
[13]
Mohammad Reza Oboudi,Energy of graphs with no eigenvalue in the interval(−1,1), Linear Algebra Appl.680(2024), 126–136
2024
-
[14]
B. R. Rakshith and L. Yashaswini,On lower bounds for the energy of graphs, Arab J. Basic Appl. Sci.33(2026), no. 1, 235–245. Seyed Ahmad Mojallal, Email:seyed_ahmad_mojallal@sfu.ca,ahmad_mojalal@yahoo.com Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada 35
2026
Reviewed August 27, 2026 · model on record in the stance chip above.
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