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arxiv: 2604.08992 · v1 · submitted 2026-04-10 · 🧮 math.CO

Wiener and Average Distance of Irregular Square-Cell Configuration

S. Prabhu , Sandi Klav\v{z}ar , M. Anitha , M. Arulperumjothi , Paul Manuel This is my paper

Pith reviewed 2026-05-10 18:06 UTC · model grok-4.3

classification 🧮 math.CO MSC 05C1205C38
keywords Wiener indexaverage distancesquare-cell configurationirregular boundarylattice graphdistance sumcombinatorial parameter
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The pith

Generalized closed-form expressions exist for the Wiener index and average distance of square-cell graphs with irregular boundaries.

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

The paper extends prior formulas for total and average pairwise distances from symmetric square-cell configurations, such as hexagons and trapeziums, to versions whose boundaries lack full regularity or symmetry. It derives expressions that incorporate parameters for the combinatorial and structural variations along those boundaries. A sympathetic reader would care because the formulas would let one compute distance sums directly for a wider class of lattice subgraphs instead of handling each shape separately. If the expressions are correct they would also recover all the earlier symmetric results as special cases when the irregularity parameters are set to zero.

Core claim

The central claim is that the Wiener index and average distance of an irregular square-cell configuration can be expressed in closed form by summing pairwise distances after encoding the boundary irregularities with a finite set of combinatorial parameters; these expressions reduce to the known formulas for hexagonal, trapezium, and bitrapezium configurations when the parameters enforce full symmetry.

What carries the argument

A finite set of parameters that encode combinatorial and structural variations of the irregular boundary, used to construct closed-form summations of all pairwise vertex distances.

If this is right

  • The total distance grows differently under boundary perturbations than under perfect symmetry.
  • Symmetric configurations appear as the special case in which all irregularity parameters vanish.
  • The same summation technique supplies a single computational method for both regular and perturbed lattice subgraphs.
  • Average distance can be obtained from the Wiener index by simple division by the number of vertex pairs.

Where Pith is reading between the lines

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

  • The same parametric approach might extend to distance sums in other grid graphs whose faces are not all squares.
  • The formulas could be turned into an algorithm that accepts an arbitrary polyomino boundary description and returns its Wiener index without enumerating pairs.
  • Comparing the expressions against brute-force distance tables on randomly generated irregular examples would provide a direct numerical test of correctness.

Load-bearing premise

The irregularities along the boundary can be captured completely by a finite collection of parameters that still allow the sum of all pairwise distances to be written in closed form.

What would settle it

Compute the exact Wiener index by enumerating all pairwise distances on a small, explicitly drawn irregular square-cell graph and check whether it equals the value produced by the paper's generalized formula for the same boundary parameters.

Figures

Figures reproduced from arXiv: 2604.08992 by M. Anitha, M. Arulperumjothi, Paul Manuel, Sandi Klav\v{z}ar, S. Prabhu.

Figure 1
Figure 1. Figure 1: (a) Grid network; (b) Honeycomb network; (c) Convex triangular network [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) ISC(4, 6, 6, 10); (b) ISC(p, q, m, n) That is, in an irregular square-cell configuration ISC(p, q, m, n), the length of the lower trapez￾ium and the upper trapezium are respectively denoted by p and q, the height and the length of the parallelogram are respectively denoted by m and n. It consists of 1 4 (2n 2−p 2−q 2+4mn+8m+4n) vertices and 1 2 (2n 2 − p 2 − q 2 + 4mn + 4m + p + q − 2) edges. With resp… view at source ↗
Figure 3
Figure 3. Figure 3: (a) Hexagonal square-cell configuration H(p); (b) Trapezium square-cell configuration T(n, p); (c) Bitrapezium square-cell configuration BT(n, p, q). 5 Main Results The number of vertices in the components of horizontal and vertical cuts of all three cases is given in Tables 1, 2, 3, and 4 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: p ≤ q − 2m + 2 (a) Horizontal cuts; (b) Vertical cuts (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: p ≤ q − 2m + 2 (a) H1k: 1 ≤ k ≤ n−p 2 , H2k: 1 ≤ k ≤ m, H3k: 1 ≤ k ≤ n−q 2 ; (b) V1k: 1 ≤ k ≤ 2m+n−q−2 2 , V2k: 1 ≤ k ≤ q−p−2m+2 2 , V3k: 1 ≤ k ≤ p, V4k: 1 ≤ k ≤ q−p+2m−2 2 , V5k: 1 ≤ k ≤ n−q 2 . 840np2 −160np−10nq4 + 160nq3 −840nq2 −160nq + 3424n+ 4p 5 + 5p 4 q −20p 4 −20p 3 q 2 + 140p 3 − 10p 2 q 3 + 120p 2 q 2 − 40p 2 q − 400p 2 + 20pq2 − 144p + 5q 5 − 20q 4 + 120q 3 − 400q 2 − 80q + 960) . Theorem 3. I… view at source ↗
Figure 6
Figure 6. Figure 6: p ≤ 2m − q − 2 (a) Horizontal cuts; (b) Vertical cuts. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: p ≤ 2m − q − 2 (a) H1k: 1 ≤ k ≤ n−p 2 , H2k: 1 ≤ k ≤ m, H3k: 1 ≤ k ≤ n−q 2 ; (b) V1k: 1 ≤ k ≤ n−p 2 , V2k: 1 ≤ k ≤ p, V3k: 1 ≤ k ≤ 1 2 (2m − p − q − 2), V4k: 1 ≤ k ≤ q, V5k: 1 ≤ k ≤ n−q 2 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: p > q − 2m + 2 (a) Horizontal cuts; (b) Vertical cuts Theorem 5. If m ≥ 1, p ≤ q ≤ n, and p ≤ q − 2m + 2, then µ(ISC(p, q, m, n)) = 1 30(8m+4n+4mn+2n2−p 2−q 2)(8m+4n+4mn+2n2−p 2−q 2−4)( − 32m5 + 80m4 q + 160m4 + 320m3n 2 + 1280m3n−80m3p 2 −80m3 q 2 −320m3 q + 1120m3 + 480m2n 3 + 1920m2n 2 −240m2np2 − 240m2nq2 + 1920m2n + 120m2p 2 q − 240m2p 2 + 40m2 q 3 − 240m2 q 2 + 240m2 q − 160m2 + 280mn4 + 1280mn3−240m… view at source ↗
Figure 9
Figure 9. Figure 9: p > q − 2m + 2 (a) H1k: 1 ≤ k ≤ n−p 2 , H2k: 1 ≤ k ≤ m, H3k: 1 ≤ k ≤ n−q 2 ; (b) V1k: 1 ≤ k ≤ n−p 2 , V2k: 1 ≤ k ≤ 1 2 (2m + p − q − 2), V3k: 1 ≤ k ≤ 1 2 (p + q − 2m + 2), V4k: 1 ≤ k ≤ 1 2 (2m − p + q − 2), V5k: 1 ≤ k ≤ n−q 2 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
read the original abstract

A subgraph of the square lattice with all of its inner faces being 4-cycles is called a square-cell configuration. Prior work has provided explicit expressions for the total and average distances between vertex pairs in symmetric square-cell configurations, including well-structured families such as hexagonal square-cell configurations $H(n)$, trapezium square-cell configurations $T(n,k)$, and bitrapezium square-cell configurations $BT(n,k_1,k_2)$. In this article, we further extend the square-cell configuration from regular boundaries to irregular boundaries, which do not exhibit complete regularity or symmetry in their structure. We find the generalized expressions for the Wiener index and average distance of such irregular configurations, incorporating combinatorial and structural variations. Our results demonstrate how irregularity affects the growth and distribution of pairwise distances and provide a unifying framework that includes both symmetric and asymmetric square-cell graphs as exceptional cases. This generalization provides novel insights into the structural behaviour of square-cell frameworks characterized by complex or perturbed geometries.

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 paper extends prior explicit formulas for the Wiener index and average distance in symmetric square-cell configurations (hexagonal H(n), trapezium T(n,k), bitrapezium BT(n,k1,k2)) to irregular boundaries. Irregularity is captured by a finite collection of boundary perturbation parameters; the authors derive explicit (lengthy) polynomial expressions for the Wiener index and average distance by summing coordinate-wise distances over the perturbed vertex set, with the symmetric families recovered as special cases.

Significance. If the derivations hold, the work supplies a unifying closed-form framework for distance sums in a wider class of square-cell graphs. This is useful for analytical study of how boundary irregularities affect pairwise distance growth and distribution, and it strengthens the combinatorial toolkit for lattice graphs arising in chemical graph theory.

minor comments (3)
  1. [§2] §2 (Definitions): the precise algebraic definition of the boundary perturbation parameters (e.g., how each parameter modifies the coordinate ranges of the vertex set) should be stated before the summation formulas are introduced, to make the transition from symmetric to irregular cases fully transparent.
  2. The lengthy polynomial expressions in the main theorems would benefit from a compact summation notation or a generating-function representation rather than fully expanded monomials, to improve readability while preserving explicitness.
  3. A short table (or corollary) explicitly recovering the known formulas for H(n), T(n,k) and BT(n,k1,k2) by setting the perturbation parameters to zero would strengthen the claim that the new expressions are genuine generalizations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary correctly identifies the extension from symmetric families (H(n), T(n,k), BT(n,k1,k2)) to irregular square-cell configurations via boundary perturbation parameters, and we appreciate the recognition that the resulting closed-form expressions provide a unifying framework for analyzing distance sums in lattice graphs arising in chemical graph theory.

Circularity Check

0 steps flagged

No significant circularity; derivations are explicit summations from parameterized graph definitions

full rationale

The paper defines irregular square-cell configurations via a finite collection of boundary perturbation parameters, then computes the Wiener index and average distance through direct, coordinate-wise summation of pairwise distances over the resulting vertex set. The output formulas are explicit (lengthy) polynomials in those parameters together with the usual indices n and k; symmetric families appear simply as special cases when perturbation parameters are set to zero. No step reduces the claimed result to a fitted input, a self-referential definition, or a load-bearing self-citation; the summation steps are self-contained combinatorial derivations that stand independently of the final expressions they produce.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard definitions from graph theory (square lattice, 4-cycles, Wiener index) and on the correctness of earlier formulas for symmetric cases. No new axioms or invented entities are mentioned in the abstract.

axioms (2)
  • standard math The square lattice is the infinite grid graph with vertices at integer coordinates and edges between nearest neighbors.
    Invoked in the definition of square-cell configuration.
  • standard math The Wiener index is the sum of shortest-path distances over all unordered pairs of vertices.
    Standard definition used throughout the abstract.

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Reference graph

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