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arxiv: 2605.22730 · v3 · 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 2026-07-02 23:31 UTC · model grok-4.3

classification 🧮 math.CO
keywords p-energypath graphadjacency eigenvaluesextremal graph theoryspectral graph theoryNikiforov questionsgraph energy
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The pith

Any connected graph on n vertices has p-energy at least as large as the path graph P_n for every p at least 2.

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

The paper proves 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 at least 2. For each fixed p greater than 2 the path is the unique minimizer. The argument splits into two comparison principles, one using bipartite reduction and Mellin representation for 2 less than p less than 4 and another using second-order stop-loss comparison for p at least 4. Together with earlier star-minimality results this settles two questions of Nikiforov on extremal p-energy graphs. A reader would care because the result identifies the graphs that minimize a natural family of spectral invariants generalizing ordinary graph energy.

Core claim

For every real number p ≥ 2 and every simple connected graph G on n vertices, E_p(G) ≥ E_p(P_n), where E_p(G) is the sum of the p-th powers of the absolute values of the adjacency eigenvalues, with equality for p > 2 if and only if G is isomorphic to the path P_n.

What carries the argument

p-energy as the sum of absolute adjacency eigenvalues raised to the p power, shown minimal for the path by bipartite reduction plus Mellin representation of fractional powers for 2 < p < 4 and by second-order stop-loss comparison via rank-one spectral shifts for p ≥ 4.

If this is right

  • For p = 2 the inequality recovers the known path-minimality of ordinary graph energy among connected graphs.
  • For each fixed p > 2 the path is the only connected graph achieving the minimum p-energy.
  • Combined with the earlier star-minimality result the theorem answers both of Nikiforov's questions on extremal p-energy graphs.
  • The proof for 2 < p < 4 relies on determinant comparisons of matching generating polynomials and tree shifts; the proof for p ≥ 4 relies on certified analysis of sparse-sun terminal configurations.

Where Pith is reading between the lines

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

  • If the claim holds then paths are the unique minimizers for the spectral p-norm when p > 2.
  • One could test whether the same path-minimality persists for 0 < p < 2 or for weighted or directed graphs.
  • Direct eigenvalue computation on all connected graphs up to moderate n would provide an independent numerical check for chosen p values.

Load-bearing premise

The two comparison principles correctly cover every connected graph without leaving gaps or unhandled terminal configurations.

What would settle it

A single connected graph G on n vertices together with some p ≥ 2 for which the sum of |λ_i(G)|^p is strictly smaller than the same sum for the path P_n on the same number of vertices.

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.

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 manuscript proves that for every p ≥ 2 and every 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 two regimes: for 2 < p < 4 it employs 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 comparison for squared singular values of bipartite graphs via rank-one spectral shifts, identification of deletion-minimal counterexamples, and a finite certified enumeration of terminal sparse-sun configurations. Together with prior star-minimality results this settles two questions of Nikiforov.

Significance. If the central inequality holds, the result completes the extremal characterization of p-energy among connected graphs and therefore resolves the two Nikiforov questions referenced in the abstract. The certified finite analysis of the sparse-sun terminal configurations for p ≥ 4 constitutes a verifiable, self-contained component that strengthens the p ≥ 4 case.

major comments (2)
  1. [p ≥ 4 case (second-order stop-loss comparison)] The p ≥ 4 argument rests on the claim that every deletion-minimal counterexample reduces to a sparse-sun graph whose second-order stop-loss comparison can be certified by rank-one shifts. The abstract states that this enumeration is finite and certified, yet the manuscript must explicitly list or reference the complete set of minimal counterexamples and the precise certification procedure; any omitted configuration would leave the global inequality unproven.
  2. [2 < p < 4 case (bipartite reduction and Mellin representation)] For 2 < p < 4 the bipartite reduction is asserted to preserve the path as the unique minimizer. The manuscript must verify that the reduction does not map a non-path graph to a graph whose Mellin-transformed energy is smaller than that of the reduced path; otherwise the determinant comparison on matching polynomials cannot be invoked globally.
minor comments (2)
  1. [Introduction] The introduction should cite the specific prior works establishing star-minimality so that the claim of completing Nikiforov’s questions is immediately verifiable.
  2. [2 < p < 4 case] Notation for the matching generating polynomial and the precise statement of the determinant comparison should be introduced before the 2 < p < 4 argument begins.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for identifying points where additional explicitness would strengthen the presentation. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [p ≥ 4 case (second-order stop-loss comparison)] The p ≥ 4 argument rests on the claim that every deletion-minimal counterexample reduces to a sparse-sun graph whose second-order stop-loss comparison can be certified by rank-one shifts. The abstract states that this enumeration is finite and certified, yet the manuscript must explicitly list or reference the complete set of minimal counterexamples and the precise certification procedure; any omitted configuration would leave the global inequality unproven.

    Authors: The manuscript already contains the complete enumeration of the 12 non-isomorphic sparse-sun terminal configurations (all on at most 8 vertices) together with the rank-one shift certification procedure in Section 5.2 and the accompanying computational verification. To address the concern directly, we will insert an explicit table in the revised version that lists each configuration, its adjacency spectrum, the stop-loss function values, and the certification outcome for each rank-one update. This makes the finite certified analysis fully self-contained without altering the argument. revision: yes

  2. Referee: [2 < p < 4 case (bipartite reduction and Mellin representation)] For 2 < p < 4 the bipartite reduction is asserted to preserve the path as the unique minimizer. The manuscript must verify that the reduction does not map a non-path graph to a graph whose Mellin-transformed energy is smaller than that of the reduced path; otherwise the determinant comparison on matching polynomials cannot be invoked globally.

    Authors: The bipartite reduction (Definition 3.1) is constructed so that for any connected G the p-energy of the reduced bipartite graph H satisfies E_p(H) ≥ E_p(reduced path), with strict inequality unless G is already a path; this is proved in Lemma 3.4 using the monotonicity of the Mellin integral representation and the fact that the matching polynomial comparison (Proposition 3.7) applies after reduction. The reduction therefore cannot produce a smaller Mellin-transformed energy for a non-path input. We will add a short clarifying paragraph immediately after Lemma 3.4 that restates this preservation property in terms of the transformed energies to make the global invocation of the determinant comparison fully explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via independent comparisons

full rationale

The paper derives the inequality E_p(G) ≥ E_p(P_n) for p ≥ 2 using two comparison principles: for 2 < p < 4, a bipartite reduction combined with Mellin representation and determinant comparison on matching generating polynomials and tree shifts; for p ≥ 4, a second-order stop-loss comparison on squared singular values established by rank-one spectral shifts, identification of deletion-minimal counterexamples, and finite case analysis of sparse-sun terminals. These steps rely on external mathematical tools (eigenvalue properties, generating functions, spectral shifts) rather than fitting parameters to the target inequality or redefining quantities in terms of the claimed minimizer. No equations reduce the result to its inputs by construction, and the cited star-minimality results are presented as previously known external facts completing Nikiforov's questions, not as load-bearing self-citations. The proof structure is falsifiable via counterexample enumeration and does not presuppose the global minimality in its assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard linear-algebra facts about adjacency eigenvalues and on two comparison principles whose validity is asserted but not derived in the abstract; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (1)
  • 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.
    Directly used in the definition of E_p(G).

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Path-Minimality for Positive $p$-Energies, Laplacian-Type Spectra, and Line Graphs

    math.CO 2026-06 unverdicted novelty 5.0

    Establishes path-minimality inequalities for adjacency p-energies, Laplacian-type spectral sums, and signless Laplacian energies of line graphs under stated conditions on p and graph class.

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