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arxiv: 2606.30996 · v1 · pith:FCLBZDSUnew · submitted 2026-06-30 · 🧮 math.CO

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

Yinchen Liu , Quanyu Tang This is my paper

Pith reviewed 2026-07-01 01:28 UTC · model grok-4.3

classification 🧮 math.CO
keywords positive p-energypath-minimalityLaplacian spectrasignless Laplacianline graphssubdivision graphsspectral sums
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The pith

Paths minimize positive p-energies among connected bipartite graphs for all real p at least 2.

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

This paper applies a companion theorem on path-minimality of adjacency p-energy to derive that the path on n vertices has the smallest positive p-energy among all connected bipartite graphs for every real p at least 2. The same minimal property holds for every connected graph when p is an odd integer at least 3 or when p equals 4. Subdivision graphs are used to obtain analogous path-minimality statements for power sums, Estrada-type indices, resolvent energies, and thresholded tails built from Laplacian and signless Laplacian spectra. A second-order stop-loss comparison for the signless Laplacian above the value 2 then yields the minimal property for the line graph of any connected graph with any number of edges.

Core claim

We prove the sharp inequality E_p^+(G) ≥ E_p^+(P_n) where P_n is the path on n vertices, in three settings: connected bipartite graphs for every real p≥2, all connected graphs for every odd integer p≥3, and all connected graphs for p=4. Using subdivision graphs, we prove path-minimality for Laplacian and signless Laplacian-type spectral sums, including power sums, Estrada-type quantities, resolvent energies, and thresholded tails. This yields the sharp line-graph inequality E_p^+(L(G)) ≥ E_p^+(P_m) for every connected graph G with m edges and every real p≥2.

What carries the argument

The path-minimality theorem for adjacency p-energy from the companion paper, transferred to positive p-energy via direct application and via subdivision graphs for Laplacian spectra.

Load-bearing premise

The path-minimality theorem for adjacency p-energy proved in the companion paper holds and applies directly to the positive p-energy settings listed here.

What would settle it

A connected bipartite graph on n vertices whose positive 2-energy is strictly smaller than the positive 2-energy of the path on n vertices would falsify the main inequality for p=2.

read the original abstract

We derive several applications of the path-minimality theorem for adjacency $p$-energy proved in the companion paper. First, we prove the sharp inequality $$ \mathcal E_p^+(G)\ge \mathcal E_p^+(P_n), $$ where $P_n$ is the path on $n$ vertices, in three settings: connected bipartite graphs for every real $p\ge2$, all connected graphs for every odd integer $p\ge3$, and all connected graphs for $p=4$. Second, using subdivision graphs, we prove path-minimality for Laplacian and signless Laplacian-type spectral sums, including power sums, Estrada-type quantities, resolvent energies, and thresholded tails. Third, we prove an edge-count second-order stop-loss comparison for the signless Laplacian above the threshold $2$. This yields the sharp line-graph inequality $$ \mathcal E_p^+(\mathcal L(G))\ge \mathcal E_p^+(P_m) $$ for every connected graph $G$ with $m$ edges and every real $p\ge2$.

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

0 major / 0 minor

Summary. The manuscript derives applications of the path-minimality theorem for adjacency p-energy from a companion paper. It proves the sharp inequality E_p^+(G) ≥ E_p^+(P_n) for connected bipartite graphs and real p≥2, for all connected graphs and odd integers p≥3, and for all connected graphs at p=4. Using subdivision graphs it establishes path-minimality for Laplacian and signless-Laplacian spectral sums (power sums, Estrada-type indices, resolvent energies, thresholded tails). It also proves an edge-count second-order stop-loss comparison for the signless Laplacian above threshold 2, yielding the sharp line-graph inequality E_p^+(L(G)) ≥ E_p^+(P_m) for every connected G with m edges and real p≥2.

Significance. If the companion theorem holds under the stated hypotheses, the results supply sharp extremal bounds for positive p-energies and extend them to Laplacian-type quantities and line graphs. The explicit matching of settings to the companion hypotheses and the use of subdivision graphs to transfer results are transparent strengths. The work adds concrete comparisons in spectral graph theory without introducing new free parameters or ad-hoc constructions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation to accept. The provided summary accurately reflects the paper's contributions and the transparent use of the companion theorem and subdivision graphs.

Circularity Check

0 steps flagged

No significant circularity; results are direct applications of external companion theorem

full rationale

The manuscript frames all central claims as applications of the path-minimality theorem for adjacency p-energy from the companion paper. The three settings for the inequality E_p^+(G) ≥ E_p^+(P_n) and the subsequent line-graph inequality are chosen to match the hypotheses of that external theorem, with no internal derivation, fitted parameters, self-definitional steps, or self-citation chains that reduce the results to inputs within this document. The derivation chain is therefore self-contained against the cited external result, which functions as independent support rather than a load-bearing self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

All claims rest on the unstated companion theorem plus standard assumptions of connectedness and the listed restrictions on p; no free parameters or invented entities appear in the abstract.

axioms (1)
  • domain assumption Path-minimality theorem for adjacency p-energy from companion paper
    All listed inequalities are derived as applications of this theorem.

pith-pipeline@v0.9.1-grok · 5719 in / 1084 out tokens · 41530 ms · 2026-07-01T01:28:55.909069+00:00 · methodology

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Reference graph

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