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arxiv: 2604.06907 · v1 · submitted 2026-04-08 · 🧮 math.CO · math.NT

Topological Indices of Divisor Prime Graphs

Purva J. Makadiya , Mahesh M. Jariya , Prashant J. Makadiya This is my paper

Pith reviewed 2026-05-10 17:50 UTC · model grok-4.3

classification 🧮 math.CO math.NT
keywords divisor prime graphtopological indicesWiener indexZagreb indicesGutman indexeccentric connectivity indexcoprime divisors
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The pith

The paper computes explicit formulas for the Wiener, Harary, Zagreb and related topological indices on the divisor prime graph.

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

The study defines the divisor prime graph G_Dp(n) whose vertices are the positive divisors of n and whose edges connect pairs of divisors that are coprime. It then derives closed expressions for eight standard topological indices, including distance-based measures such as the Wiener and hyper-Wiener indices and degree-based measures such as the first and second Zagreb indices. A reader would care if these formulas convert questions about graph distances and degrees into direct arithmetic statements about the divisors of n. If the derivations hold, the indices become computable from the divisor lattice alone without enumerating edges or paths for each n.

Core claim

For any positive integer n let D(n) be its set of positive divisors. The graph G_Dp(n) has vertex set D(n) and places an edge between distinct divisors x and y precisely when gcd(x,y)=1. On this graph the authors obtain explicit formulas for the Wiener index, Harary index, hyper-Wiener index, first Zagreb index, second Zagreb index, Schultz index, Gutman index and eccentric connectivity index, each expressed in terms of sums or counts taken over the elements of D(n).

What carries the argument

The divisor prime graph G_Dp(n), whose adjacency rule (edges exactly when gcd=1) turns the usual divisor set into a concrete graph on which the standard definitions of Wiener-type and Zagreb-type indices can be evaluated directly.

If this is right

  • The Wiener index of G_Dp(n) equals a specific sum over pairs of coprime divisors weighted by their graph distance.
  • The first and second Zagreb indices reduce to sums of squares of the numbers of divisors coprime to each given divisor.
  • All listed indices become functions of the prime factorization of n once the formulas are applied.
  • The eccentric connectivity index depends only on the maximum distances from each divisor to the farthest coprime divisor.

Where Pith is reading between the lines

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

  • If the indices turn out to be multiplicative or otherwise factor nicely with n, they could serve as new invariants for distinguishing numbers with the same divisor count.
  • The same adjacency rule might be applied to other arithmetic sets, such as the set of integers up to n, to produce comparable families of indices.
  • Closed forms for these indices open the possibility of studying their asymptotic growth as n tends to infinity through its divisors.

Load-bearing premise

That applying the usual topological-index definitions to this coprimality graph on divisors automatically produces arithmetic information that is both new and useful.

What would settle it

For a concrete small n such as 12, compute the Wiener index by enumerating all pairwise distances in G_Dp(12) and check whether the number equals the closed formula given in the paper.

read the original abstract

Graph theory provides powerful tools for modeling concepts in number theory, leading to the introduction of graphs derived from arithmetic properties. One such structure is the divisor prime graph, $G_{Dp(n)}$. For any positive integer $n$, let $D(n)$ be the set of its positive divisors. The vertex set of $G_{Dp(n)}$ consists of the elements of $D(n)$, with the adjacency condition that two vertices $x$ and $y$ share an edge if and only if their greatest common divisor is $1$. The primary focus of this study is to evaluate the topological characteristics of $G_{Dp(n)}$. To achieve this, we analyze and compute various distance and degree-based indices, specifically focusing on the Wiener, Harary, hyper-Wiener, First and Second Zagreb, Schultz, Gutman, and Eccentric connectivity 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

0 major / 3 minor

Summary. The manuscript defines the divisor prime graph G_{Dp(n)} on the set D(n) of positive divisors of n, with an edge between x and y precisely when gcd(x,y)=1. It then derives explicit expressions for the Wiener, Harary, hyper-Wiener, first and second Zagreb, Schultz, Gutman, and eccentric-connectivity indices of this graph.

Significance. If the derivations are correct, the work supplies closed-form formulas that express standard topological indices directly in terms of the divisor set of n. This constitutes a routine but non-trivial enumeration over the divisor poset and may be of modest interest to researchers working at the intersection of number theory and chemical graph theory, particularly for quick evaluation of these invariants on arithmetic graphs.

minor comments (3)
  1. The abstract states that the indices are computed but supplies neither the explicit formulas nor any verification steps; the main body should present the derivations in a self-contained manner with at least one worked example for a small n (e.g., n=6 or n=30) to allow immediate checking.
  2. Notation for the graph is introduced as G_Dp(n) in the title and G_{Dp(n)} in the abstract; adopt a single consistent LaTeX form throughout.
  3. The paper would benefit from a brief comparison, even qualitative, of the obtained index values with those of the ordinary divisor graph or the coprimality graph on {1,...,n} to situate the results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of our manuscript and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript defines G_Dp(n) explicitly via the coprimality adjacency rule on the divisor set D(n) and then applies the standard, externally defined formulas for the listed topological indices (Wiener index as sum of pairwise distances, Zagreb indices as sums over vertex degrees, etc.). These steps are direct, non-referential computations from the graph's adjacency matrix and distance matrix; no parameter is fitted and then relabeled as a prediction, no self-citation supplies a load-bearing uniqueness or ansatz, and no known result is merely renamed. The derivation chain therefore remains self-contained against the explicit graph construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definitions of divisors, gcd, and the listed topological indices; no free parameters or new entities are introduced beyond the graph definition itself.

axioms (2)
  • domain assumption The positive divisors D(n) of any positive integer n form a well-defined finite set that can serve as the vertex set of a graph.
    Invoked in the opening definition of G_Dp(n).
  • domain assumption Two divisors x and y are adjacent precisely when gcd(x,y)=1.
    The adjacency rule that defines the graph.

pith-pipeline@v0.9.0 · 5451 in / 1255 out tokens · 30683 ms · 2026-05-10T17:50:54.681846+00:00 · methodology

discussion (0)

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Reference graph

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9 extracted references · 9 canonical work pages

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