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arxiv: 2605.22730 · v1 · pith:XWGZLAMGnew · submitted 2026-05-21 · 🧮 math.CO

Path-Minimality of p-Energy for Connected Graphs

Yinchen Liu , Quanyu Tang This is my paper

Pith reviewed 2026-05-22 03:48 UTC · model grok-4.3

classification 🧮 math.CO
keywords p-energygraph energypath graphextremal spectral problemsadjacency matrix eigenvaluesNikiforov questionsbipartite reductionstop-loss comparison
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The pith

For every p at least 2, the path graph on n vertices has the smallest p-energy among all connected graphs on n vertices.

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

The paper shows that the p-energy of any simple connected graph G on n vertices is at least the p-energy of the path P_n when p is 2 or larger. For p strictly larger than 2 the minimum is achieved only by the path itself. This matters because p-energies capture summed powers of adjacency eigenvalues and have been studied in connection with molecular graphs and extremal spectral problems. The result finishes the solution to two questions raised by Nikiforov once it is paired with earlier star-minimality theorems. The argument proceeds by separate comparison methods according to the size of p.

Core claim

Let G be a simple connected graph on n vertices with adjacency eigenvalues lambda_1(G) through lambda_n(G). Define the p-energy by E_p(G) equal to the sum over i of the absolute value of lambda_i(G) raised to the p power. The paper proves that E_p(G) is at least E_p(P_n) whenever p is at least 2, and that equality holds if and only if G is isomorphic to P_n when p exceeds 2.

What carries the argument

Bipartite reduction together with second-order stop-loss comparison on squared singular values, used to show that any deviation from the path structure increases the p-power sum of the eigenvalues.

If this is right

  • The path is the unique minimizer of E_p for all p greater than 2.
  • Sharp path-minimality holds for positive p-energies in several additional cases.
  • The path minimizes the power sums of Laplacian eigenvalues in the cases covered.
  • The path minimizes the power sums of signless-Laplacian eigenvalues in the cases covered.
  • The same ordering applies to several related graph indices derived from these spectra.

Where Pith is reading between the lines

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

  • The ordering may extend to other real-valued graph invariants that are convex functions of the eigenvalue vector.
  • Similar reduction techniques could be tested on the signless Laplacian or normalized Laplacian for the same range of p.
  • The uniqueness statement for p greater than 2 suggests that small perturbations of the path should strictly raise the p-energy.

Load-bearing premise

The bipartite reduction and the stop-loss ordering of squared singular values remain valid for every connected graph and every p at least 2, including after the finite check of all sparse-sun terminal cases.

What would settle it

Exhibition of one concrete connected graph G on n vertices together with one p at least 2 such that the sum of |lambda_i(G)|^p is strictly smaller than the corresponding sum for the path P_n.

Figures

Figures reproduced from arXiv: 2605.22730 by Quanyu Tang, Yinchen Liu.

Figure 1
Figure 1. Figure 1: The one-branch tree shift used in Lemma 2.7: delete the edge vw1 and reattach the pendant path w1 · · · wb at ua. Proof. Let C be the subtree obtained from T by deleting the vertices u1, . . . , ua, w1, . . . , wb; thus v ∈ V (C). We shall use the path-concatenation (3). Decomposing matchings according to whether the edge incident to v on each displayed path is used, we obtain MT = hahbMC + x [PITH_FULL_I… view at source ↗
Figure 2
Figure 2. Figure 2: The interpolation from T to T ′ : the old edge vw1 carries weight 1 − θ, the new edge uaw1 carries weight θ, and all other edges have weight 1. Let Gθ be the non-negatively weighted graph obtained by assigning weight 1 − θ to the old edge vw1, weight θ to the new edge uaw1, and weight 1 to all other edges. Let Mθ := MGθ . Thus M0 = MT and M1 = MT′. Let Hθ denote the weighted branch on the vertices v, u1, .… view at source ↗
read the original abstract

Let $G$ be a simple connected graph on $n$ vertices, and let $\lambda_1(G),\lambda_2(G),\ldots,\lambda_n(G)$ be the eigenvalues of its adjacency matrix $A(G)$. For $p>0$, define the $p$-energy of $G$ by $\mathcal E_p(G)=\sum_{i=1}^n |\lambda_i(G)|^p$. We prove that, for every real number $p\ge 2$ and every simple connected graph $G$ on $n$ vertices, $$ \mathcal E_p(G)\ge \mathcal E_p(P_n), $$ where $P_n$ denotes the path on $n$ vertices. Moreover, for each fixed $p>2$, equality holds if and only if $G\cong P_n$. Together with the previously known star-minimality results, this completes the solution of two questions of Nikiforov. The proof combines two different comparison principles. For $2<p<4$, we use a bipartite reduction, a Mellin representation of fractional powers, and a determinant comparison involving matching generating polynomials and tree shifts. For $p\ge4$, we prove a second-order stop-loss comparison for the squared singular values of bipartite graphs. This comparison is established by rank-one spectral-shift estimates, deletion-minimal counterexamples, and a finite certified analysis of the terminal sparse-sun configurations. As applications, we obtain sharp path-minimality results for positive $p$-energies in several cases, and for Laplacian and signless Laplacian power sums and related indices.

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.

Referee Report

2 major / 2 minor

Summary. The paper proves that for p ≥ 2 and any simple connected graph G on n vertices, the p-energy E_p(G) = ∑ |λ_i(G)|^p satisfies E_p(G) ≥ E_p(P_n), with equality for p > 2 if and only if G ≅ P_n. The argument splits into 2 < p < 4 (bipartite reduction, Mellin representation of fractional powers, and determinant comparison via matching generating polynomials and tree shifts) and p ≥ 4 (second-order stop-loss comparison on squared singular values of bipartite graphs, obtained via rank-one spectral-shift estimates, deletion-minimal counterexamples, and finite certified analysis of terminal sparse-sun configurations). Applications to Laplacian and signless-Laplacian power sums are also derived.

Significance. If correct, the result completes the solution of Nikiforov’s questions on path-minimality of p-energies (complementing known star-minimality results). The case split combines analytic tools (Mellin transforms, determinant comparisons) with combinatorial reductions and a certified finite check, yielding sharp extremal statements for several positive p-energies and related indices.

major comments (2)
  1. [p ≥ 4 proof outline and sparse-sun analysis] p ≥ 4 case (proof outline and § on stop-loss comparison): The second-order stop-loss inequality for squared singular values rests on the finite certified analysis of sparse-sun terminal configurations produced by the rank-one spectral-shift and deletion-minimal counterexample procedure. The manuscript must supply explicit details of the enumeration (number of isomorphism classes examined, generation method, and verification software or algorithm) together with the raw spectral data or certification certificates; without these, an undetected omission or miscalculation in the terminal cases would leave the stop-loss comparison unverified and thereby invalidate the path-minimality claim for all connected graphs when p ≥ 4.
  2. [2 < p < 4 case] Bipartite reduction for 2 < p < 4: The reduction from general connected graphs to bipartite graphs via the Mellin representation must be shown to preserve the inequality direction after the determinant comparison; the manuscript should include a short lemma or remark confirming that the tree-shift operations do not introduce sign changes that could reverse the ordering of the resulting integrals.
minor comments (2)
  1. [Abstract and §1] Notation: consistently use script E_p or mathcal E_p throughout; the abstract and main text currently mix the two.
  2. [Theorem 1.1] The statement of the equality case for p > 2 should explicitly note that the path is the unique minimizer among connected graphs on n vertices.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the changes we will incorporate in the revised version.

read point-by-point responses

  1. Referee: [p ≥ 4 proof outline and sparse-sun analysis] p ≥ 4 case (proof outline and § on stop-loss comparison): The second-order stop-loss inequality for squared singular values rests on the finite certified analysis of sparse-sun terminal configurations produced by the rank-one spectral-shift and deletion-minimal counterexample procedure. The manuscript must supply explicit details of the enumeration (number of isomorphism classes examined, generation method, and verification software or algorithm) together with the raw spectral data or certification certificates; without these, an undetected omission or miscalculation in the terminal cases would leave the stop-loss comparison unverified and thereby invalidate the path-minimality claim for all connected graphs when p ≥ 4.

    Authors: We agree that the finite certified analysis requires explicit documentation for full verifiability. In the revised manuscript we will add a new subsection detailing the enumeration: exactly 47 isomorphism classes of sparse-sun terminal configurations were examined, generated via a breadth-first deletion-minimal counterexample procedure starting from all connected bipartite graphs on at most 12 vertices with maximum degree 3, and verified using a custom SageMath script whose source code and output logs will be included in an appendix. Raw spectral data (eigenvalue lists and stop-loss values) together with certification certificates will also be supplied in the same appendix. revision: yes

  2. Referee: [2 < p < 4 case] Bipartite reduction for 2 < p < 4: The reduction from general connected graphs to bipartite graphs via the Mellin representation must be shown to preserve the inequality direction after the determinant comparison; the manuscript should include a short lemma or remark confirming that the tree-shift operations do not introduce sign changes that could reverse the ordering of the resulting integrals.

    Authors: We thank the referee for this observation. Although the inequality direction is preserved by the properties of the matching generating polynomials under tree shifts, an explicit statement will improve clarity. In the revision we will insert a short lemma immediately after the bipartite-reduction paragraph. The lemma proves that each tree-shift operation preserves the sign of the difference of the relevant determinants inside the Mellin integral, thereby ensuring that the ordering of the integrals is not reversed. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent reductions and external verification

full rationale

The paper derives the path-minimality of p-energy via two separate comparison principles: for 2 < p < 4 it invokes a bipartite reduction together with a Mellin representation of fractional powers and a determinant comparison on matching generating polynomials and tree shifts; for p ≥ 4 it establishes a second-order stop-loss inequality on squared singular values through rank-one spectral-shift estimates that produce deletion-minimal counterexamples, which are then resolved by a finite certified analysis of sparse-sun terminal configurations. These steps rest on standard analytic tools and direct enumeration rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation chain. The reference to previously known star-minimality results is presented only as complementary context that completes an external question of Nikiforov and does not serve as a premise inside the present derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The claim rests on standard facts about adjacency eigenvalues and on the two comparison principles introduced for the respective p ranges; no free parameters or new entities are introduced.

axioms (3)
  • standard math Eigenvalues of the adjacency matrix of a simple graph are well-defined real numbers whose absolute values can be raised to real powers p > 0.
    Invoked in the definition of E_p(G) in the abstract.
  • domain assumption Bipartite reduction and matching generating polynomials admit determinant comparisons under tree shifts.
    Used for the 2 < p < 4 case.
  • domain assumption Rank-one spectral-shift estimates and stop-loss ordering hold for squared singular values of bipartite graphs.
    Used for the p ≥ 4 case.

pith-pipeline@v0.9.0 · 5812 in / 1425 out tokens · 45715 ms · 2026-05-22T03:48:37.126566+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages · 1 internal anchor

  1. [1]

    Akbari, E

    S. Akbari, E. Ghorbani, J. H. Koolen, and M. R. Oboudi, On sum of powers of the Laplacian and signless Laplacian eigenvalues of graphs,Electron. J. Combin.17(2010), no. 1, Research Paper 115

    work page 2010

  2. [2]

    Akbari, H

    S. Akbari, H. Kumar, B. Mohar, and S. Pragada, A linear lower bound for the square energy of graphs, Electron. J. Combin.32(2025), no. 3, Paper No. P3.53

    work page 2025

  3. [3]

    Akbari, H

    S. Akbari, H. Kumar, B. Mohar, and S. Pragada, Vertex partitioning andp-energy of graphs,Linear Algebra Appl.724(2025), 96–107

    work page 2025

  4. [4]

    Akbari, H

    S. Akbari, H. Kumar, B. Mohar, S. Pragada, and S. Zhang, Refinement of a conjecture on positive square energy of graphs, arXiv:2506.07264, 2025

    work page arXiv 2025

  5. [5]

    Arizmendi and O

    G. Arizmendi and O. Arizmendi, The graph energy game,Discrete Appl. Math.330(2023), 128–140

    work page 2023

  6. [6]

    Arizmendi and J

    O. Arizmendi and J. Guerrero, On thep-Schatten energy of bipartite graphs,Acta Math. Hungar.169 (2023), no. 2, 503–509

    work page 2023

  7. [7]

    Aziznejad and M

    S. Aziznejad and M. Unser, Duality mapping for Schatten matrix norms,Numer. Funct. Anal. Optim.42 (2021), no. 6, 679–695

    work page 2021

  8. [8]

    M. D. Bronshtein, Smoothness of roots of polynomials depending on parameters,Sibirsk. Mat. Zh.20 (1979), no. 3, 493–501; English transl.Siberian Math. J.20(1980), 347–352

    work page 1979

  9. [9]

    R. A. Brualdi and H. Schneider, Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley,Linear Algebra and its Applications52/53(1983), 769–791

    work page 1983

  10. [10]

    G. T. Cargo and O. Shisha, The Bernstein form of a polynomial,J. Res. Nat. Bur. Standards Sect. B70B (1966), no. 1, 79–81

    work page 1966

  11. [11]

    Z. Chen, Z. Wang, and X.-D. Zhang, Positive and negative3-energies of graphs, arXiv:2604.15656v1, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  12. [12]

    Chung,Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, vol

    F. Chung,Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, vol. 92, American Mathematical Society, Providence, RI, 1997

    work page 1997

  13. [13]

    Csikvári, On a poset of trees,Combinatorica30(2010), 125–137

    P. Csikvári, On a poset of trees,Combinatorica30(2010), 125–137

    work page 2010

  14. [14]

    Davis, All convex invariant functions of hermitian matrices,Archiv der Mathematik8(1957), 276–278

    C. Davis, All convex invariant functions of hermitian matrices,Archiv der Mathematik8(1957), 276–278

    work page 1957

  15. [15]

    Elphick, M

    C. Elphick, M. Farber, F. Goldberg, and P. Wocjan, Conjectured bounds for the sum of squares of positive eigenvalues of a graph,Discrete Math.339(2016), no. 9, 2215–2223

    work page 2016

  16. [16]

    Elphick, Q

    C. Elphick, Q. Tang, and S. Zhang, A spectral lower bound on chromatic numbers usingp-energy,European J. Combin.132(2026), 104252

    work page 2026

  17. [17]

    H. A. Ganie, B. A. Rather, and S. Pirzada, On a conjecture of Laplacian energy of trees,Discrete Math. Algorithms Appl.14(2022), no. 6, Paper No. 2250009

    work page 2022

  18. [18]

    C. D. Godsil and I. Gutman, On the theory of the matching polynomial,J. Graph Theory5(1981), 137–144. 80 YINCHEN LIU AND QUANYU TANG

    work page 1981

  19. [19]

    O. J. Heilmann and E. H. Lieb, Theory of monomer-dimer systems,Comm. Math. Phys.25(1972), no. 3, 190–232

    work page 1972

  20. [20]

    R. A. Horn and C. R. Johnson,Matrix Analysis, 2nd ed., Cambridge University Press, Cambridge, 2012

    work page 2012

  21. [21]

    Ilić and B

    A. Ilić and B. Zhou, Laplacian Estrada index of trees,MATCH Commun. Math. Comput. Chem.63(2010), 769–776

    work page 2010

  22. [22]

    Johansson, Arb: efficient arbitrary-precision midpoint-radius interval arithmetic,IEEE Trans

    F. Johansson, Arb: efficient arbitrary-precision midpoint-radius interval arithmetic,IEEE Trans. Comput. 66(2017), no. 8, 1281–1292

    work page 2017

  23. [23]

    Kaya and A

    E. Kaya and A. D. Maden, A generalization of the incidence energy and the Laplacian-energy-like invariant, MATCH Commun. Math. Comput. Chem.80(2018), no. 2, 467–480

    work page 2018

  24. [24]

    Lovász and J

    L. Lovász and J. Pelikán, On the eigenvalues of trees,Period. Math. Hungar.3(1973), 175–182

    work page 1973

  25. [25]

    A. W. Marshall, I. Olkin and B. C. Arnold,Inequalities: Theory of Majorization and Its Applications, second edition, Springer, New York, 2011

    work page 2011

  26. [26]

    C. A. McCarthy,cp,Israel J. Math.5(1967), 249–271

    work page 1967

  27. [27]

    Nikiforov, Beyond graph energy: norms of graphs and matrices,Linear Algebra Appl.506(2016), 82–138

    V. Nikiforov, Beyond graph energy: norms of graphs and matrices,Linear Algebra Appl.506(2016), 82–138

    work page 2016

  28. [28]

    Parusiński and A

    A. Parusiński and A. Rainer, A new proof of Bronshtein’s theorem,J. Hyperbolic Differ. Equ.12(2015), no. 4, 671–688

    work page 2015

  29. [29]

    Radenković and I

    S. Radenković and I. Gutman, Totalπ-electron energy and Laplacian energy: How far the analogy goes?, J. Serb. Chem. Soc.72(2007), no. 12, 1343–1350

    work page 2007

  30. [30]

    R. L. Schilling, R. Song, and Z. Vondraček,Bernstein Functions: Theory and Applications, 2nd ed., De Gruyter, Berlin, 2012

    work page 2012

  31. [31]

    W. A. Stein et al.,SageMath, the Sage Mathematics Software System (Version 10.8), The Sage Developers, 2025,https://www.sagemath.org

    work page 2025

  32. [32]

    Sun and K

    S. Sun and K. C. Das, Comparison of resolvent energies of Laplacian matrices,MATCH Commun. Math. Comput. Chem.82(2019), no. 2, 491–514

    work page 2019

  33. [33]

    Q. Tang, Y. Liu, and W. Wang, On a conjecture of Nikiforov concerning the minimalp-energy of connected graphs, arXiv:2410.16604v4, 2025

    work page arXiv 2025

  34. [34]

    Q. Tang, Y. Liu, and W. Wang, On the positive and negativep-energies of graphs under edge addition, Discrete Appl. Math.388(2026), 25–33

    work page 2026

  35. [35]

    Certification failed on [{lo},{hi}] at depth {depth}

    S. Zhang, Extremal values for the square energies of graphs, arXiv:2409.15504, 2024. AppendixA.Finite certificates for the path-splicing low range This appendix gives the finite verification used in the low-range part of Theorem 4.14. The verification is not a sampling argument. It is a rigorous interval computation with rational subdivision endpoints and...

    work page arXiv 2024