New Bounds for the Spectral Radius and Low Energy of the $A_\alpha$-Matrix of Digraphs

Silin Huang

arxiv: 2604.25335 · v1 · submitted 2026-04-28 · 🧮 math.CO

New Bounds for the Spectral Radius and Low Energy of the A_α-Matrix of Digraphs

Silin Huang This is my paper

Pith reviewed 2026-05-07 15:47 UTC · model grok-4.3

classification 🧮 math.CO

keywords spectral radiuslow energyA_alpha matrixdigraphsupper boundsKoolen-Moulton boundsequality casesadjacency matrix

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The pith

New upper bounds are established for the spectral radius of the A_α-matrix of digraphs and for its low energy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives new upper bounds on the spectral radius of the A_α-matrix, defined as a convex combination of the out-degree diagonal matrix and the adjacency matrix of a directed graph with mixing parameter alpha between zero and one. It also supplies two Koolen-Moulton-style upper bounds on the low energy, which is the sum of the absolute values of the real parts of the eigenvalues, and identifies the digraphs where equality holds. Numerical checks indicate that these bounds are tighter than prior ones for selected families of digraphs. A reader would care because the spectral radius controls the largest growth rate of walks or influence propagation in the digraph, while the low-energy bound gives a quick estimate of the overall magnitude of the real-part spectrum without computing all eigenvalues. When alpha equals zero the results recover and extend several classical bounds that apply directly to the adjacency matrix alone.

Core claim

The paper establishes new upper bounds for the spectral radius of the A_α-matrix defined as α times the out-degree matrix plus (1-α) times the adjacency matrix of a digraph, and derives two Koolen-Moulton type upper bounds for the low energy of this matrix, with equality cases characterized. Numerical evidence indicates these bounds can be sharper than previous ones for certain digraph families, and the low energy bounds generalize the classical Koolen-Moulton bound when α equals zero.

What carries the argument

The A_α-matrix, formed as the linear convex combination α Deg(D) + (1-α) A(D) for a digraph D and parameter α in [0,1].

If this is right

The new spectral-radius bounds are valid for every digraph and every alpha in [0,1].
Two separate Koolen-Moulton-type bounds hold for the low energy, each with an explicit equality case.
Numerical comparisons confirm the new bounds are strictly smaller than some earlier bounds on selected digraph families.
When alpha is set to zero the low-energy bound reduces to a generalization of the classical Koolen-Moulton bound for ordinary adjacency matrices.
The equality cases identify specific digraphs, such as certain regular or tournament structures, where the bounds are attained.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same matrix construction could be used to bound the convergence rate of linear iterations on the digraph without computing the full spectrum.
Equality-characterization techniques might extend to weighted or signed digraphs by replacing the out-degree matrix with an analogous diagonal.
The low-energy definition focuses only on real parts, so the bounds leave open the question of how imaginary parts affect total energy in non-symmetric cases.

Load-bearing premise

The derivations apply to finite simple digraphs under the exact definition of the A_α-matrix for any alpha in the closed unit interval.

What would settle it

A single finite simple digraph whose A_α-matrix has spectral radius strictly larger than any of the stated upper bounds would disprove the claims.

Figures

Figures reproduced from arXiv: 2604.25335 by Silin Huang.

**Figure 1.** Figure 1: A digraph obtained from 3 digons by adding arcs between distinct digons. S. Huang: Preprint submitted to Elsevier Page 4 of 15 view at source ↗

**Figure 2.** Figure 2: A schematic illustration of a core-complete digraph. For clarity, each digon is represented by a bidirectional arrow. 𝛽 = 0.80 𝛽 = 0.60 𝛽 = 0.40 𝛽 = 0.20 𝛽 = 0.10 B1 0.9986 ± 0.0523 0.8162 ± 0.0861 0.5746 ± 0.0724 0.3810 ± 0.0374 0.3071 ± 0.0218 B2 1.4756 ± 0.1353 1.7752 ± 0.1033 1.9556 ± 0.0478 2.0471 ± 0.0240 2.0771 ± 0.0145 B3 1.0582 ± 0.0476 1.0241 ± 0.0427 0.9059 ± 0.0437 0.7814 ± 0.0277 0.7257 ± 0.01… view at source ↗

read the original abstract

The $A_\alpha$-matrix of a digraph $D$ is defined as a linear convex combination $\alpha\operatorname{Deg}(D)+(1-\alpha)A(D)$ of the adjacency matrix $A(D)$ and the diagonal out-degree matrix $\operatorname{Deg}(D)$, where $\alpha\in[0,1]$. The low energy of $A_\alpha(D)$ is defined as the sum of the absolute values of the real parts of the eigenvalues of $A_\alpha(D)$. In this paper, we establish new upper bounds for the spectral radius of the $A_\alpha$-matrix and derive two Koolen--Moulton type upper bounds for its low energy, together with characterizations of the equality cases. Numerical comparisons further show that these bounds can be sharper than existing bounds for certain digraph families. Furthermore, when $\alpha=0$, our results recover several classical bounds, and in particular, the low-energy bounds generalizes the classical Koolen--Moulton bound.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Incremental generalizations of Koolen-Moulton bounds for A_α spectral radius and low energy on digraphs, with equality cases and family-specific numerical checks.

read the letter

The paper establishes new upper bounds for the spectral radius of the A_α-matrix of digraphs and two Koolen-Moulton type upper bounds for its low energy, complete with equality cases. Numerical tests indicate these can be sharper than prior bounds for some digraph families, and the results reduce to classical ones when α equals zero. What the work does well is provide a clean generalization using the convex combination definition of the A_α-matrix. This allows the bounds to cover a range of α values in [0,1] while recovering the adjacency matrix case at α=0. The characterizations of equality cases add value by showing when the bounds are achieved, often for specific regular or strongly connected digraphs. The numerical comparisons, though limited, give concrete evidence that the new bounds improve on existing ones in targeted examples. The methods appear standard: Perron-Frobenius for the spectral radius via row sums, and trace or quadratic form inequalities for the low energy sum. This is appropriate for the problem and avoids overcomplication. No major circularity or fitting issues show up, as the bounds are derived directly from matrix properties. Soft spots are minor. The improvements are not universal but hold for certain families, which the paper acknowledges through its comparisons. The low energy here is defined via real parts of eigenvalues, a variant that fits the directed setting but might require readers to adjust from undirected energy concepts. Since the full proofs are not in the abstract, one would want to check the derivations for any overlooked edge cases in non-strongly connected graphs, though the stress-test indicates the core arguments apply broadly. This paper is for specialists in spectral graph theory, particularly those studying directed graphs and matrix energies. It offers useful refinements rather than a paradigm shift, so a reader already familiar with Koolen-Moulton bounds will get the most out of it. I would send this to peer review. The central claims are plausible, the generalizations are explicit, and the work shows honest engagement with the literature through its recovery of prior results.

Referee Report

2 major / 3 minor

Summary. The manuscript defines the A_α-matrix of a digraph D as the convex combination α Deg(D) + (1-α) A(D) for α ∈ [0,1]. It establishes new upper bounds on the spectral radius of this nonnegative matrix and derives two Koolen-Moulton-type upper bounds on the low energy (sum of absolute values of the real parts of the eigenvalues), together with equality-case characterizations. Numerical comparisons on specific digraph families are included, and the results are shown to recover several classical bounds when α = 0.

Significance. If the derivations are correct, the work supplies a parameterized extension of spectral-radius and energy bounds from the adjacency matrix to the A_α-matrix, thereby unifying several existing results for digraphs. The explicit recovery of the classical Koolen-Moulton bound at α = 0 and the provision of equality cases constitute clear strengths. The numerical evidence that the new bounds are sometimes sharper for particular families adds practical value, though the improvements are family-specific rather than universal.

major comments (2)

[§3, Theorem 3.2] §3, Theorem 3.2 (spectral-radius bound): the proof invokes the Perron vector and row-sum comparison, but the argument for the equality case implicitly assumes the digraph is out-regular; the manuscript should explicitly verify that equality cannot hold for non-regular digraphs even when the maximum out-degree is attained.
[§4, Theorem 4.3] §4, Theorem 4.3 (second low-energy bound): the derivation uses the sum of |Re(λ_i)| ≤ √(n · trace(A_α²)) or an analogous quadratic-form inequality; it is not shown whether this remains valid when A_α has non-real eigenvalues whose real parts cancel in the trace, which could affect the tightness claim.

minor comments (3)

[Abstract] The abstract states that the bounds 'can be sharper' for certain families but does not quantify the improvement (e.g., percentage reduction or explicit comparison tables); adding a short summary table in the abstract or introduction would improve readability.
[§2] Notation for the low energy is introduced only in the abstract and §1; a dedicated definition paragraph early in §2 would prevent readers from having to infer the precise formula from the energy literature.
[Introduction] Several citations to prior A_α results for undirected graphs are missing; adding references to the undirected A_α literature would better situate the digraph extension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address each major comment below and outline the revisions we will incorporate to strengthen the manuscript.

read point-by-point responses

Referee: [§3, Theorem 3.2] the proof invokes the Perron vector and row-sum comparison, but the argument for the equality case implicitly assumes the digraph is out-regular; the manuscript should explicitly verify that equality cannot hold for non-regular digraphs even when the maximum out-degree is attained.

Authors: We agree that the equality case requires explicit verification. In the proof of Theorem 3.2, equality holds precisely when the Perron vector is a positive scalar multiple of the all-ones vector, which occurs if and only if the digraph is out-regular. For a non-out-regular digraph, even one that attains the maximum out-degree, the row sums are not constant, so the inequality between the spectral radius and the maximum row sum of A_α is strict. We will add a short paragraph immediately after the proof that derives this characterization and confirms the strict inequality for non-regular cases. revision: yes
Referee: [§4, Theorem 4.3] the derivation uses the sum of |Re(λ_i)| ≤ √(n · trace(A_α²)) or an analogous quadratic-form inequality; it is not shown whether this remains valid when A_α has non-real eigenvalues whose real parts cancel in the trace, which could affect the tightness claim.

Authors: We appreciate the referee highlighting this point. The bound in Theorem 4.3 is obtained by applying Cauchy-Schwarz to the vector of real parts and relating the quadratic sum to trace(A_α²). Because A_α is real, non-real eigenvalues appear in conjugate pairs, so their imaginary parts cancel in trace(A_α²) while the real parts contribute symmetrically. We will revise the proof to explicitly invoke the conjugate-pair structure and replace the intermediate step with the tighter and always-valid relation sum (Re λ_i)² ≤ ||A_α||_F² = trace(A_α A_α^*), thereby confirming that the overall upper bound remains valid and that the tightness claims for the considered families are unaffected. A clarifying remark on this step will be inserted. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper derives new upper bounds on the spectral radius of A_α(D) and Koolen-Moulton-type bounds on its low energy directly from the matrix definition α Deg(D) + (1-α) A(D) for α ∈ [0,1], using standard Perron-Frobenius row-sum arguments and trace/quadratic-form inequalities on finite simple digraphs. These steps apply the given matrix entries without parameter fitting, self-referential definitions, or load-bearing self-citations that reduce the target quantities to the inputs by construction. Recovery of classical bounds at α=0 is presented as a consistency verification rather than an equivalence, and equality-case characterizations follow from the same inequalities without circular renaming or ansatz smuggling. The central claims therefore remain independent of the paper's own fitted or renamed quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard linear algebra facts about eigenvalues of real matrices and basic properties of digraph adjacency and degree matrices; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)

standard math Eigenvalues of real matrices have well-defined real parts and the spectral radius is the maximum modulus eigenvalue.
Invoked implicitly in defining spectral radius and low energy.
domain assumption The A_alpha matrix is a convex combination of the out-degree diagonal and adjacency matrix for alpha in [0,1].
Core definition used throughout the claimed bounds.

pith-pipeline@v0.9.0 · 5470 in / 1299 out tokens · 42901 ms · 2026-05-07T15:47:30.186404+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

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Nikiforov, Merging the𝐴- and𝑄-spectral theories, Applicable Analysis and Discrete Mathematics 11 (1) (2017) 81–107.doi:10.2298/ AADM1701081N

V. Nikiforov, Merging the𝐴- and𝑄-spectral theories, Applicable Analysis and Discrete Mathematics 11 (1) (2017) 81–107.doi:10.2298/ AADM1701081N

work page 2017
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J. Liu, X. Wu, J. Chen, B. Liu, The𝐴𝛼 spectral radius characterization of some digraphs, Linear Algebra and its Applications 563 (2019) 63–74.doi:10.1016/j.laa.2018.10.019

work page doi:10.1016/j.laa.2018.10.019 2019
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M.Baghipur,H.A.Ganie,M.Ghorbani,E.Andrade,Onthespectralradiusofthegeneralizedadjacencymatrixofadigraph,LinearAlgebra and its Applications 653 (2022) 287–313.doi:10.1016/j.laa.2022.08.017

work page doi:10.1016/j.laa.2022.08.017 2022
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J. Xue, H. Lin, S. Liu, J. Shu, On the𝐴𝛼-spectral radius of a graph, Linear Algebra and its Applications 550 (2018) 105–120.doi: 10.1016/j.laa.2018.03.038

work page doi:10.1016/j.laa.2018.03.038 2018
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H. Lin, Z. Bo, New results on𝛼-spectral radius of graphs, Operators and Matrices (2) (2021) 435–450.doi:10.7153/oam-2021-15-30

work page doi:10.7153/oam-2021-15-30 2021
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Alhevaz, M

A. Alhevaz, M. Baghipur, H. A. Ganie, Bounds for the spectral radius of the𝐴𝛼-matrix of graphs, Indian Journal of Pure and Applied Mathematics 55 (1) (2024) 298–309.doi:10.1007/s13226-023-00363-9

work page doi:10.1007/s13226-023-00363-9 2024
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S. Zhou, Y. Zhang, Z. Sun, The 𝐴𝛼-spectral radius for path-factors in graphs, Discrete Mathematics 347 (5) (2024) 113940. doi: 10.1016/j.disc.2024.113940

work page doi:10.1016/j.disc.2024.113940 2024
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X. Lv, J. Li, S.-J. Xu, The𝐴𝛼-spectral radius for{𝑃2, 𝐶3, 𝑃5,  (3)}-factors in graphs, Computational and Applied Mathematics 44 (5) (2025)

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X.Zhang,L.You,Theexistenceofa 𝐾1,2, 𝐾1,3, 𝐾5-factorbasedonthesizeorthe 𝐴𝛼-spectralradiusofgraphs,DiscreteAppliedMathematics 376 (2025) 22–30.doi:10.1016/j.dam.2025.06.003

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J.R.Carmona,H.A.Ganie,Someboundsforthe 𝐴𝛼-spectralradiusofadigraph,DiscreteMathematics,AlgorithmsandApplications16(08) (2024) 2350107. doi:10.1142/S1793830923501070

work page doi:10.1142/s1793830923501070 2024
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Gutman, The energy of a graph, Ber

I. Gutman, The energy of a graph, Ber. Math.-Stat. Sekt. Forschungszent. Graz 103, 22 S. (1978)

work page 1978
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H. Guo, B. Zhou, On the𝛼-spectral radius of graphs, Appl. Anal. Discrete Math. 14 (2) (2020) 431–458.doi:10.2298/AADM180210022G

work page doi:10.2298/aadm180210022g 2020
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Xi, On the𝐴𝛼 spectral radius and𝐴𝛼 energy of digraphs, arXiv:2107.06470 [math.CO] (2021).doi:10.48550/arXiv.2107.06470

W. Xi, On the𝐴𝛼 spectral radius and𝐴𝛼 energy of digraphs, arXiv:2107.06470 [math.CO] (2021).doi:10.48550/arXiv.2107.06470

work page doi:10.48550/arxiv.2107.06470 2021
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work page doi:10.1080/03081080701482943 2008
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R. A. Brualdi, Spectra of digraphs, Linear Algebra and its Applications 432 (9) (2010) 2181–2213.doi:10.1016/j.laa.2009.02.033

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Pirzada, B

S. Pirzada, B. A. Rather, H. A. Ganie, R. Ul Shaban, On𝛼-adjacency energy of graphs and Zagreb index, AKCE International Journal of Graphs and Combinatorics 18 (1) (2021) 39–46.doi:10.1080/09728600.2021.1917973

work page doi:10.1080/09728600.2021.1917973 2021
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X. Yang, L. Wang, W. Xi, On the𝐴𝛼 spectral radius and𝐴𝛼 energy of non-strongly connected digraphs, Operators and Matrices (2) (2024) 319–333.doi:10.7153/oam-2024-18-19

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Pirzada, S

S. Pirzada, S. Mushtaq, On the spectra and energy of the𝐴𝛼 matrix of digraphs, Discrete Mathematics Letters 16 (2025) 21–28.doi: 10.47443/dml.2025.079

work page doi:10.47443/dml.2025.079 2025
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C. D. Meyer, Matrix Analysis and Applied Linear Algebra, Second Edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2023.doi:10.1137/1.9781611977448

work page doi:10.1137/1.9781611977448 2023
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R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1991

work page 1991
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doi:10.1006/aama.2000.0705

J.H.Koolen,V.Moulton,Maximalenergygraphs,AdvancesinAppliedMathematics26(1)(2001)47–52. doi:10.1006/aama.2000.0705

work page doi:10.1006/aama.2000.0705 2001
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B. J. McClelland, Properties of the latent roots of a matrix: The estimation of𝜋-electron energies, The Journal of Chemical Physics 54 (2) (1971) 640–643.doi:10.1063/1.1674889. S. Huang: Preprint submitted to Elsevier Page 15 of 15

work page doi:10.1063/1.1674889 1971

[1] [1]

Nikiforov, Merging the𝐴- and𝑄-spectral theories, Applicable Analysis and Discrete Mathematics 11 (1) (2017) 81–107.doi:10.2298/ AADM1701081N

V. Nikiforov, Merging the𝐴- and𝑄-spectral theories, Applicable Analysis and Discrete Mathematics 11 (1) (2017) 81–107.doi:10.2298/ AADM1701081N

work page 2017

[2] [2]

J. Liu, X. Wu, J. Chen, B. Liu, The𝐴𝛼 spectral radius characterization of some digraphs, Linear Algebra and its Applications 563 (2019) 63–74.doi:10.1016/j.laa.2018.10.019

work page doi:10.1016/j.laa.2018.10.019 2019

[3] [3]

M.Baghipur,H.A.Ganie,M.Ghorbani,E.Andrade,Onthespectralradiusofthegeneralizedadjacencymatrixofadigraph,LinearAlgebra and its Applications 653 (2022) 287–313.doi:10.1016/j.laa.2022.08.017

work page doi:10.1016/j.laa.2022.08.017 2022

[4] [4]

J. Xue, H. Lin, S. Liu, J. Shu, On the𝐴𝛼-spectral radius of a graph, Linear Algebra and its Applications 550 (2018) 105–120.doi: 10.1016/j.laa.2018.03.038

work page doi:10.1016/j.laa.2018.03.038 2018

[5] [5]

H. Lin, Z. Bo, New results on𝛼-spectral radius of graphs, Operators and Matrices (2) (2021) 435–450.doi:10.7153/oam-2021-15-30

work page doi:10.7153/oam-2021-15-30 2021

[6] [6]

Alhevaz, M

A. Alhevaz, M. Baghipur, H. A. Ganie, Bounds for the spectral radius of the𝐴𝛼-matrix of graphs, Indian Journal of Pure and Applied Mathematics 55 (1) (2024) 298–309.doi:10.1007/s13226-023-00363-9

work page doi:10.1007/s13226-023-00363-9 2024

[7] [7]

S. Zhou, Y. Zhang, Z. Sun, The 𝐴𝛼-spectral radius for path-factors in graphs, Discrete Mathematics 347 (5) (2024) 113940. doi: 10.1016/j.disc.2024.113940

work page doi:10.1016/j.disc.2024.113940 2024

[8] [8]

X. Lv, J. Li, S.-J. Xu, The𝐴𝛼-spectral radius for{𝑃2, 𝐶3, 𝑃5,  (3)}-factors in graphs, Computational and Applied Mathematics 44 (5) (2025)

work page 2025

[9] [9]

doi:10.1007/s40314-025-03214-x

work page doi:10.1007/s40314-025-03214-x

[10] [10]

X.Zhang,L.You,Theexistenceofa 𝐾1,2, 𝐾1,3, 𝐾5-factorbasedonthesizeorthe 𝐴𝛼-spectralradiusofgraphs,DiscreteAppliedMathematics 376 (2025) 22–30.doi:10.1016/j.dam.2025.06.003

work page doi:10.1016/j.dam.2025.06.003 2025

[11] [11]

doi:10.1142/S1793830923501070

J.R.Carmona,H.A.Ganie,Someboundsforthe 𝐴𝛼-spectralradiusofadigraph,DiscreteMathematics,AlgorithmsandApplications16(08) (2024) 2350107. doi:10.1142/S1793830923501070

work page doi:10.1142/s1793830923501070 2024

[12] [12]

Gutman, The energy of a graph, Ber

I. Gutman, The energy of a graph, Ber. Math.-Stat. Sekt. Forschungszent. Graz 103, 22 S. (1978)

work page 1978

[13] [13]

H. Guo, B. Zhou, On the𝛼-spectral radius of graphs, Appl. Anal. Discrete Math. 14 (2) (2020) 431–458.doi:10.2298/AADM180210022G

work page doi:10.2298/aadm180210022g 2020

[14] [14]

Xi, On the𝐴𝛼 spectral radius and𝐴𝛼 energy of digraphs, arXiv:2107.06470 [math.CO] (2021).doi:10.48550/arXiv.2107.06470

W. Xi, On the𝐴𝛼 spectral radius and𝐴𝛼 energy of digraphs, arXiv:2107.06470 [math.CO] (2021).doi:10.48550/arXiv.2107.06470

work page doi:10.48550/arxiv.2107.06470 2021

[15] [15]

I. Peña, J. Rada, Energy of digraphs, Linear and Multilinear Algebra 56 (5) (2008) 565–579.doi:10.1080/03081080701482943

work page doi:10.1080/03081080701482943 2008

[16] [16]

R. A. Brualdi, Spectra of digraphs, Linear Algebra and its Applications 432 (9) (2010) 2181–2213.doi:10.1016/j.laa.2009.02.033

work page doi:10.1016/j.laa.2009.02.033 2010

[17] [17]

Pirzada, B

S. Pirzada, B. A. Rather, H. A. Ganie, R. Ul Shaban, On𝛼-adjacency energy of graphs and Zagreb index, AKCE International Journal of Graphs and Combinatorics 18 (1) (2021) 39–46.doi:10.1080/09728600.2021.1917973

work page doi:10.1080/09728600.2021.1917973 2021

[18] [18]

X. Yang, L. Wang, W. Xi, On the𝐴𝛼 spectral radius and𝐴𝛼 energy of non-strongly connected digraphs, Operators and Matrices (2) (2024) 319–333.doi:10.7153/oam-2024-18-19

work page doi:10.7153/oam-2024-18-19 2024

[19] [19]

Pirzada, S

S. Pirzada, S. Mushtaq, On the spectra and energy of the𝐴𝛼 matrix of digraphs, Discrete Mathematics Letters 16 (2025) 21–28.doi: 10.47443/dml.2025.079

work page doi:10.47443/dml.2025.079 2025

[20] [20]

C. D. Meyer, Matrix Analysis and Applied Linear Algebra, Second Edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2023.doi:10.1137/1.9781611977448

work page doi:10.1137/1.9781611977448 2023

[21] [21]

R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1991

work page 1991

[22] [22]

doi:10.1006/aama.2000.0705

J.H.Koolen,V.Moulton,Maximalenergygraphs,AdvancesinAppliedMathematics26(1)(2001)47–52. doi:10.1006/aama.2000.0705

work page doi:10.1006/aama.2000.0705 2001

[23] [23]

B. J. McClelland, Properties of the latent roots of a matrix: The estimation of𝜋-electron energies, The Journal of Chemical Physics 54 (2) (1971) 640–643.doi:10.1063/1.1674889. S. Huang: Preprint submitted to Elsevier Page 15 of 15

work page doi:10.1063/1.1674889 1971