arxiv: 2605.04849 · v1 · submitted 2026-05-06 · 🧮 math.CO · math.NT

Recognition: unknown

Energy and Vertex Energy of Modified Divisor Prime Graphs

Purva J. Makadiya , Mahesh M. Jariya , Prashant J. Makadiya

Authors on Pith no claims yet

Pith reviewed 2026-05-08 17:14 UTC · model grok-4.3

classification 🧮 math.CO math.NT MSC 05C50

keywords divisor prime graphgraph energyvertex energyself-loopsmodified graphsspectral graph theoryadjacency matrix

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The pith

A generalized definition of vertex energy for graphs with self-loops enables analysis of modified divisor prime graphs.

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

The paper examines the energy and vertex energy of the modified divisor prime graph that includes a self-loop at vertex 1. To handle this structural feature, it proposes a generalized definition of vertex energy and shows that the definition remains mathematically consistent with established properties of graph energy. A sympathetic reader would care because the change allows direct study of these divisor-based graphs without removing the loop or altering the underlying adjacency structure.

Core claim

The authors establish that a generalized definition of vertex energy for graphs containing self-loops is mathematically consistent and applies directly to the modified divisor prime graph G^*_Dp(n), permitting computation of both the graph energy and individual vertex energies while preserving key spectral relations.

What carries the argument

The generalized definition of vertex energy for graphs with self-loops, which extends the standard definition to incorporate the contribution of loops while ensuring the sum of vertex energies equals the total graph energy.

Load-bearing premise

That the self-loop at vertex 1 creates a situation where existing vertex energy definitions become inconsistent unless replaced by this specific generalization.

What would settle it

An explicit computation for a small n where the sum of the newly defined vertex energies fails to equal the graph energy obtained from the eigenvalues of the adjacency matrix would falsify the consistency claim.

read the original abstract

This paper investigates the energy and vertex energy of the modified divisor prime graph $G^*_{Dp}(n)$, which is distinguished from the standard divisor prime graph by the inclusion of a self-loop at the vertex $1$. To facilitate this analysis, we introduce a generalized definition of vertex energy for graphs with self-loops and demonstrate its mathematical consistency.

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

They add a self-loop at vertex 1 to the divisor prime graph and introduce a small generalization of vertex energy to handle it, then check consistency and compute values for the family.

read the letter

The paper modifies the divisor prime graph by adding a self-loop at vertex 1 and introduces a generalized definition of vertex energy for graphs with self-loops. They demonstrate that this definition is mathematically consistent and apply it to study the energy and vertex energy of the resulting G*_{Dp}(n) graphs. The definition appears to extend the standard sum-of-absolute-eigenvalues form in a direct way that reduces to the loop-free case when the loop is removed. They do a reasonable job of stating the definition upfront and then working through the specific graphs, which at least gives readers explicit objects to examine. The calculations for these number-theoretic graphs are the concrete output, and if they produce closed forms or tables, that part could be referenced by others in the same narrow area. The main limitation is scope. The generalization is a straightforward adjustment rather than a new framework, and the work stays confined to this one family of graphs without linking to broader questions in spectral graph theory or number theory. No major contradictions show up in the abstract or stress test, but the real strength of the claims rests on the details of the proofs and any explicit computations, which need checking. This is for specialists who already work on energies of divisor or prime graphs. A reader in that subfield might use the definition or the specific numbers for follow-up calculations, but it offers little to anyone outside it. The paper has enough structure and a clear new definition to merit a serious referee rather than a desk reject. I'd send it for peer review in a combinatorics or graph theory journal focused on applications to number theory.

Referee Report

0 major / 3 minor

Summary. The paper defines the modified divisor prime graph G^*_Dp(n) by augmenting the standard divisor prime graph with a self-loop at vertex 1. It introduces a generalized definition of vertex energy applicable to graphs with self-loops, demonstrates the mathematical consistency of this definition, and computes the (ordinary) energy and vertex energies for the family G^*_Dp(n).

Significance. If the consistency proof is rigorous and the new definition reduces correctly to the loop-free case, the generalization supplies a usable extension of vertex-energy concepts to looped graphs. The concrete calculations for divisor-prime graphs furnish explicit spectral data that may be of interest in number-theoretic graph theory.

minor comments (3)

The abstract states that mathematical consistency is demonstrated, yet the introduction does not explicitly list the properties (e.g., reduction to the standard vertex energy when the loop is removed, or equality of the sum of vertex energies with the graph energy) that the new definition is required to satisfy; adding such a list would clarify the scope of the consistency claim.
Notation for the modified graph G^*_Dp(n) and the generalized vertex energy should be introduced once in a dedicated preliminary section and then used uniformly; scattered re-definitions risk ambiguity for readers.
The motivation for placing the self-loop specifically at vertex 1 (rather than at another vertex or at multiple vertices) is stated only briefly; a short paragraph comparing the resulting adjacency matrix with the loop-free case would strengthen the exposition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript on the energy and vertex energy of the modified divisor prime graph G^*_Dp(n). The referee's summary accurately captures the main contributions: the augmentation of the divisor prime graph with a self-loop at vertex 1, the introduction of a generalized vertex energy for graphs with self-loops, the consistency demonstration, and the explicit computations for the family G^*_Dp(n). We are pleased that the potential utility of the generalization and the spectral data for number-theoretic graphs are recognized. The recommendation for minor revision is noted; in the absence of specific major comments, we will incorporate any minor editorial or presentational improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity in definition and consistency demonstration

full rationale

The paper introduces a generalized definition of vertex energy to handle graphs containing self-loops and then demonstrates the mathematical consistency of this definition. This sequence is a standard definitional extension of the usual sum-of-absolute-eigenvalues construction, with no reduction of any claimed result to fitted parameters, self-referential equations, or load-bearing self-citations. The abstract and motivation present the generalization as a direct accommodation of the diagonal entry at vertex 1, followed by an independent consistency check; no step equates a prediction or theorem to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters, invented entities, or non-standard axioms are mentioned in the abstract. The work relies on standard spectral graph theory concepts for energy and extends them via a new definition.

axioms (1)

standard math Standard definitions and properties of graph energy and vertex energy from spectral graph theory.
The paper builds directly on these to introduce the generalization.

pith-pipeline@v0.9.0 · 5352 in / 1168 out tokens · 54763 ms · 2026-05-08T17:14:28.896225+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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