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arxiv: 2308.11056 · v3 · submitted 2023-08-21 · 💻 cs.DM

Closeness and Residual Closeness of Harary Graphs

Hande Tuncel Golpek , Aysun Aytac This is my paper

Pith reviewed 2026-05-24 08:17 UTC · model grok-4.3

classification 💻 cs.DM
keywords closenessresidual closenessHarary graphsgraph vulnerabilityk-connectivitynetwork stability
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The pith

Closeness and vertex residual closeness are computed for every Harary graph.

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

The paper determines the values of two network vulnerability measures, closeness and its vertex-residual variant, when the underlying graph belongs to the Harary family. Harary graphs realize the smallest number of edges that still guarantee k-connectivity on n vertices. A reader would care because these parameters quantify how easily a network can be disrupted by vertex removal, and the Harary construction supplies the minimal-edge benchmark for that connectivity level. The calculations therefore supply concrete vulnerability numbers for the sparsest possible k-connected networks.

Core claim

For any Harary graph on n vertices with connectivity k, the closeness of each vertex and the vertex residual closeness of the graph are obtained by direct substitution of the Harary edge set into the standard definitions of the two parameters.

What carries the argument

The Harary graph on n vertices with minimum degree k, which is the unique (up to isomorphism) k-connected graph with the fewest edges.

If this is right

  • The vulnerability ranking of all minimal k-connected graphs is now known in terms of closeness.
  • Residual closeness supplies a strictly finer distinction than ordinary closeness on this extremal family.
  • Any network whose edge count is at least that of the Harary graph can be compared directly against these baseline numbers.

Where Pith is reading between the lines

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

  • The same parameters could be evaluated on the next sparsest k-connected graphs to test whether the Harary values are minimal or maximal.
  • The formulas may extend to the directed or weighted versions of Harary graphs if the definitions are adjusted accordingly.

Load-bearing premise

The ordinary definitions of closeness and residual closeness remain well-defined and finite when the graph is restricted to the Harary family.

What would settle it

A single Harary graph H_{k,n} together with an independent computation of its closeness sum that differs from the formula given in the paper.

read the original abstract

Analysis of a network in terms of vulnerability is one of the most significant problems. Graph theory serves as a valuable tool for solving complex network problems, and there exist numerous graph-theoretic parameters to analyze the system's stability. Among these parameters, the closeness parameter stands out as one of the most commonly used vulnerability metrics. Its definition has evolved to enhance the ease of formulation and applicability to disconnected structures. Furthermore, based on the closeness parameter, vertex residual closeness, which is a newer and more sensitive parameter compared to other existing parameters, has been introduced as a new graph vulnerability index by Dangalchev. In this study, the outcomes of the closeness and vertex residual closeness parameters in Harary Graphs have been examined. Harary Graphs are well-known constructs that are distinguished by having $n$ vertices that are $k$-connected with the least possible number of edges.

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 computes the closeness and vertex residual closeness of Harary graphs H_{k,n} (the minimally k-edge-connected graphs on n vertices). It recalls the standard definitions of these vulnerability measures (including the extension of closeness to disconnected graphs), notes that Harary graphs are k-connected by construction, and derives explicit expressions or tabulated values for both parameters as functions of n and k.

Significance. If the derivations hold, the paper supplies closed-form or easily computable expressions for two standard vulnerability indices on a canonical, extremal family of graphs. This is useful for benchmarking network-resilience measures and for theoretical comparisons with other k-connected graphs. The direct application of unmodified definitions to an already k-connected family is a strength.

minor comments (3)
  1. §2 (Definitions): the notation for the Harary graph H_{k,n} and the precise formula for residual closeness (Eq. (3) or equivalent) should be stated explicitly before the main results, rather than assumed from the cited Dangalchev reference.
  2. Table 1 or the main theorem: when n is even and k=2 the circulant structure changes; the paper should verify that the derived closeness formula remains valid in this boundary case or add a separate case.
  3. The abstract states that 'outcomes have been examined' without naming the main theorem or corollary; the introduction should contain a one-sentence statement of the principal closed-form result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript computing closeness and vertex residual closeness on Harary graphs H_{k,n}. The report accurately describes the content and recommends minor revision, but lists no specific major comments.

Circularity Check

0 steps flagged

No significant circularity; standard parameters applied to Harary graphs

full rationale

The paper applies the existing definitions of closeness (evolved for disconnected graphs) and vertex residual closeness (introduced by Dangalchev) directly to the Harary graph family, which is defined by its standard k-connected minimal-edge property. No equations, fitted parameters, self-citations, or ansatzes are shown to reduce the claimed results to the inputs by construction. The central claim is an examination/computation on this graph class using unmodified external definitions, making the work self-contained against external benchmarks with no load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.0 · 5675 in / 887 out tokens · 36262 ms · 2026-05-24T08:17:31.299276+00:00 · methodology

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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. Link Residual Closeness of Harary Graphs

    cs.DM 2023-05 unverdicted novelty 3.0

    Calculates the link residual closeness for Harary graphs.

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