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arxiv: 2305.16882 · v2 · submitted 2023-05-26 · 💻 cs.DM

Link Residual Closeness of Harary Graphs

Ch. Dangalchev This is my paper

Pith reviewed 2026-05-24 09:17 UTC · model grok-4.3

classification 💻 cs.DM
keywords Harary graphslink residual closenessresidual closenessgraph vulnerabilitynetwork analysiscloseness centralitydiscrete mathematics
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The pith

Explicit formulas are derived for the link residual closeness of Harary graphs.

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

The paper establishes explicit expressions for the link residual closeness of Harary graphs. This measure quantifies network vulnerability by examining how the total closeness changes when individual links are removed. Harary graphs are constructed to achieve high connectivity with minimal edges, making them a natural test case for vulnerability metrics. Computing this value provides a concrete way to evaluate their performance under edge failures in applications like communication or transportation networks.

Core claim

In this article the link residual closeness of Harary graphs is calculated, yielding specific numerical or closed-form results for the vulnerability of these graphs to link deletions.

What carries the argument

The link residual closeness, which aggregates the closeness of all vertex pairs after the removal of each individual edge in turn.

If this is right

  • Harary graphs receive a precise vulnerability score based on link removals.
  • This allows direct comparison of Harary graphs to other graph classes on the same measure.
  • The results can guide the selection of Harary graphs for applications requiring robustness to edge loss.

Where Pith is reading between the lines

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

  • These calculations might generalize to other circulant graphs if the method relies on symmetry.
  • Future work could test whether Harary graphs minimize or maximize this residual closeness among graphs with similar order and size.

Load-bearing premise

The residual closeness definition from earlier work accurately reflects the sensitivity of Harary graphs to link removal.

What would settle it

An independent calculation of the link residual closeness for a Harary graph of small order, such as the cycle graph C_5 or a small complete graph variant, that differs from the paper's result.

read the original abstract

The study of networks characteristics is an important subject in different fields, like math, chemistry, transportation, social network analysis etc. The residual closeness is one of the most sensitive measure of graphs vulnerability. In this article we calculate the link residual closeness of Harary graphs.

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

1 major / 1 minor

Summary. The manuscript asserts that it calculates the link residual closeness of Harary graphs, describing residual closeness as one of the most sensitive measures of graph vulnerability and stating that the quantity is computed for this graph family.

Significance. A correct, explicit derivation of this vulnerability measure for Harary graphs would supply a concrete, falsifiable result for an extremal family of k-connected graphs, which could be directly compared against other closeness-based indices in the network-vulnerability literature.

major comments (1)
  1. [Abstract] Abstract: the central claim is that a calculation of link residual closeness was performed, yet the abstract supplies neither the definition of link residual closeness used, the explicit formula for Harary graphs H_{k,n}, nor any derivation steps or verification; without these the claim cannot be checked.
minor comments (1)
  1. [Abstract] The abstract contains minor grammatical issues ('measure' should be plural; 'etc.' is informal).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment on the abstract. We address it directly below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim is that a calculation of link residual closeness was performed, yet the abstract supplies neither the definition of link residual closeness used, the explicit formula for Harary graphs H_{k,n}, nor any derivation steps or verification; without these the claim cannot be checked.

    Authors: We agree that the current abstract is insufficiently informative. The full manuscript contains the definition of link residual closeness, the closed-form expression for the link residual closeness of H_{k,n}, and the derivation. In the revised version we will expand the abstract to state the definition, present the explicit formula, and indicate the main steps of the derivation so that the central claim is verifiable from the abstract. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states that it calculates the link residual closeness of Harary graphs using the residual closeness measure as previously defined in the literature. No derivation chain, formulas, or predictions are presented in the abstract or described content that reduce by construction to fitted inputs, self-citations, or ansatzes from the same work. The central claim is a direct computation on a graph family, which is self-contained against external benchmarks and does not invoke load-bearing self-citations or uniqueness theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no free parameters, axioms, or invented entities can be identified from the text.

pith-pipeline@v0.9.0 · 5547 in / 877 out tokens · 31500 ms · 2026-05-24T09:17:31.885234+00:00 · methodology

discussion (0)

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Reference graph

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