Distance Sequences to bound the Harary Index and other Wiener-type Indices of a Graph

Peter Dankelmann

REVIEW 3 major objections 6 minor 44 references

Distance Sequences to bound the Harary Index and other Wiener-type Indices of a Graph

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Distance-sequence maximality makes one graph extremal for every monotone Wiener-type index and resolves the Harary-index open problem.

desk verdict A clean distance-sequence framework that gives sharp Harary/hyper-Wiener bounds, but the load-bearing lemmas are imported from earlier work and one theorem has a sign typo. read the letter →

arxiv 2411.13439 v1 pith:OZFNDNGB submitted 2024-11-20 math.CO

Peter Dankelmann This is my paper · ORCID

classification math.CO MSC 05C1205C3505C4005C10

keywords generalisedWienerindexHararyhyper-Wienermultiplicativedistancesequenceextremalgraphspath-completegraph

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's central claim is that many extremal questions for distance-based graph indices can be settled at once by comparing a single object, the distance sequence: the sorted list of all pairwise shortest-path distances. If one graph's list is at least as large as another's entry by entry, then every nondecreasing index, such as the hyper-Wiener index, is at least as large, and every nonincreasing index, such as the Harary index, is at least as small. The paper identifies, for several graph classes, a graph whose distance sequence dominates all others, and thereby derives sharp extremal bounds for every monotone distance-based index at once. In particular, among connected graphs of fixed order and size, the path-complete graph has minimum Harary index and maximum hyper-Wiener index, resolving an open problem from the monograph on the Harary index. The same template covers even-connectivity graphs, maximal k-degenerate graphs, maximal outerplanar graphs, Apollonian networks, and trees with all degrees odd.

What carries the argument

The central object is the distance sequence $D(G)$, the nondecreasing list of distances between all unordered pairs of vertices, compared entry by entry. The transfer statement, Proposition 1, converts a distance-sequence comparison into an index comparison for any index that is monotone in each coordinate; this is why all named indices can be handled at once. The work in each section is to find a graph whose distance sequence dominates every graph in the class: the path-complete graph $PK_{n,m}$ for fixed order and size, the $\kappa/2$-th power of the cycle $C_n^{\kappa/2}$ for even connectivity $\kappa$, the $k$-th power of the path $P_n^k$ for maximal $k$-degenerate graphs, and the odd tree $T_n$ for trees with all degrees odd. Lemma 1, the deletion inequality $D(G) \le D(G-v) \odot D_G(v)$ for a non-cut vertex $v$, is the inductive engine that proves the new distance-sequence extrema.

What would settle it

Enumerate all connected graphs with n vertices and m edges for n up to 7 and compare, entry by entry, their sorted distance lists with the distance list of the path-complete graph; finding any graph whose list is larger in some coordinate would disprove Theorem 1. The same exhaustive check against the relevant power of the cycle for 2-connected and 4-connected graphs would test the even-connectivity theorem.

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Extended reading notes

Core claim

The discovery is that the extremal graph for a whole family of distance-based indices is determined by the maximum of the distance sequence, not by the particular index. A distance-based index is any function of the sorted distance list $D(G)$, and it is nondecreasing or nonincreasing according to how it responds to each entry. The paper's transfer principle says: if every graph in a class satisfies $D(G) \le D(H^*)$ for a fixed graph $H^*$, then $H^*$ maximises every nondecreasing distance-based index and minimises every nonincreasing one. Taking $H^*$ to be the path-complete graph $PK_{n,m}$, a path whose end vertex is joined to a clique, gives the main theorem: for connected graphs of order $n$ and size $m$, $PK_{n,m}$ maximises the hyper-Wiener and multiplicative Wiener indices and minimises the Harary index, resolving the open problem from the Harary-index monograph. The same transfer applied to known or newly proved distance-sequence maxima yields $C_n^{\kappa/2}$ for even-$\kappa$-connected graphs, $P_n^k$ for maximal $k$-degenerate graphs and $k$-trees, and $T_n$ for odd trees.

Load-bearing premise

The whole argument depends on two maximum-distance-sequence facts imported from an earlier paper: every connected graph with n vertices and m edges has distance sequence no larger than that of the path-complete graph, and every even-connectivity graph has distance sequence no larger than that of the appropriate power of a cycle; if either imported fact is false, the corresponding sharp bounds, including the Harary-index resolution, collapse.

Editorial extensions

If this is right

Among all connected graphs of order $n$ and size $m$, the path-complete graph simultaneously minimises the Harary index and maximises the hyper-Wiener and multiplicative Wiener indices, settling the open problem from the Harary-index monograph.
For every even $\kappa$, the graph $C_n^{\kappa/2}$ is extremal for all nondecreasing distance-based indices among $\kappa$-connected graphs of order $n$, and for $\kappa = 2$ the cycle is the unique extremal graph.
Among maximal $k$-degenerate graphs and $k$-trees, $P_n^k$ is extremal for all nondecreasing distance-based indices; this makes $P_n^2$ extremal for maximal outerplanar graphs and $P_n^3$ for Apollonian networks.
Among trees whose vertices all have odd degree, the tree $T_n$ is extremal for all nondecreasing distance-based indices.
Any future monotone distance-based index automatically inherits these same extremal graphs, because extremality is a property of the distance sequence rather than of the individual index.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If a distance-sequence maximum is found for odd connectivity, the same template would immediately yield sharp Harary and hyper-Wiener bounds for $\kappa$-connected graphs with odd $\kappa$, a case the paper leaves open.
The paper leaves open whether all maximal planar graphs have distance sequence at most that of $P_n^3$; a positive answer would give sharp Harary and hyper-Wiener bounds for all maximal planar graphs by the same argument.
Computationally, the approach reduces bounding any monotone distance-based index on these classes to evaluating one fixed sequence, so closed-form or algorithmic bounds follow directly from the explicit distance sequence of the extremal graph.
Because the extremality criterion is the distance sequence, the same machinery could be reused in any graph class where a dominating distance sequence is known or can be constructed, such as classes with prescribed diameter or degree sequence.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does two useful things. First, it formalizes a simple and transferable idea: if you know a graph maximizes the distance sequence (componentwise) in a class, then every nondecreasing distance-based index is maximized by that same graph. Proposition 1 plus the path-complete graph's distance-sequence maximality from Soltés/Casablanca–Dankelmann gives an immediate sharp upper bound on the hyper-Wiener index and lower bound on the Harary index for fixed order and size, resolving an explicit open problem in Xu–Das–Trinajstić. That is a real result, even if the derivation is short. Second, the paper extends the trick to new classes with self-contained proofs: maximal k-degenerate graphs (Lemma 5) and odd trees (Lemma 6). The proofs of Lemmas 5 and 6 are structurally sound; I checked the induction steps and the sequence bounds. The fact that the same extremal graph works for all monotone indices, not just Wiener, is a genuine consolidation.

The soft spots are minor and mostly editorial. Theorem 2(b) states the wrong direction: for κ=2 and an increasing index, the inequality should be I(G) ≤ I(C_n), not ≥. As written it contradicts Theorem 2(a). It is almost certainly a typo, but it needs fixing. Corollary 1 has a duplicated label: the second part should be (b), not (a). Lemma 6's proof has a small garbled phrase ("the neighbour w of v in T′") that should read "of v1 in T′." None of these affect the main arguments.

The more substantive concern is the reliance on Lemmas 2 and 3, both imported without proof from [11]. The central Theorem 1 is just Proposition 1 plus Lemma 2, so the paper's headline result inherits all of its weight from a cited lemma. That is normal mathematical practice, but I would want a referee to verify that [11] indeed proves componentwise distance-sequence maximality of PK_{n,m} and C_n^{κ/2}, and not merely Wiener maximality. The stress-test note is right to flag this. It is not a reason to reject; it is a reason to have a careful referee check the cited lemmas.

Who is this for? People working in chemical graph theory and distance-based indices will get immediate use from the Harary and hyper-Wiener bounds. The framework is simple enough to be taught. I would send it to a serious referee. After fixing the typo, the duplication, and verifying the two imported lemmas, I would accept it.

Recommendation: send to peer review with a referee who can check [11].

Referee Report

3 major / 6 minor

Summary. The paper develops a unified method for bounding distance-based topological indices by comparing distance sequences. It shows that if a graph class has a member G* with componentwise maximal distance sequence, then every nondecreasing distance-based index is maximized by G*, and every nonincreasing one is minimized. The paper identifies such extremal graphs for four classes: the path-complete graph for connected graphs of given order and size (Theorem 1), powers of cycles for even-vertex-connectivity (Theorem 2), powers of paths for maximal k-degenerate graphs and k-trees (Theorem 3 and Corollary 1), and a particular tree T_n for odd trees (Theorem 4). Applications include a sharp lower bound on the Harary index among graphs of given order and size, which resolves an open problem from Xu, Das, and Trinajstić's monograph, and sharp upper bounds on the hyper-Wiener and multiplicative Wiener indices. Lemmas 5 and 6 are proved in the paper; Lemmas 2 and 3 are cited from the author's earlier paper [11].

Significance. The distance-sequence framework is a clean and productive idea: one extremal distance-sequence result yields bounds for an entire family of indices (Wiener, Harary, hyper-Wiener, multiplicative Wiener, variable Wiener, etc.) simultaneously. The resolution of the Harary-index open problem is a concrete contribution, and the results for maximal outerplanar graphs, Apollonian networks, and odd trees extend the literature in classes where only Wiener-index results were known. The in-paper proofs (Lemmas 5 and 6) are elementary and appear sound. The principal risks are the two imported lemmas and a false inequality in Theorem 2(b); both are local and repairable, so the paper is likely suitable for publication after revision.

major comments (3)

[Section 4, Lemma 2 and Theorem 1] Theorem 1 is a direct corollary of Lemma 2, but Lemma 2 is imported from [11] without proof or a verbatim statement of the result in [11]. Componentwise maximality of D(PK_{n,m}) is strictly stronger than Wiener-index maximality (the sum of the entries), so a proof that PK_{n,m} maximizes the Wiener index alone would not suffice. Since [11] is about strong products and the current paper does not reproduce the argument, the central claim resolving the Harary-index problem cannot be verified from the submitted text. The author should include a proof of Lemma 2 or quote the exact theorem from [11] with all hypotheses.
[Section 5, Lemma 3 and Theorem 2] Theorem 2 depends entirely on Lemma 3, also cited from [11] without proof. The same request applies: either reproduce the proof or state the precise theorem from [11] that establishes componentwise maximality of D(C_n^{κ/2}) among κ-connected graphs. This is load-bearing for the Harary and hyper-Wiener bounds for even connectivity.
[Section 5, Theorem 2(b)] The inequality as printed is false. For an increasing index I, Lemma 3 and Proposition 1 give I(G) ≤ I(C_n^{κ/2}), not ≥. For instance, the Wiener index is increasing, and among 2-connected graphs on four vertices, W(K_4)=6 < W(C_4)=8. The inequality should read ≤, and the equality statement 'iff G is a cycle' is then consistent for strictly increasing indices.

minor comments (6)

[Section 6, Corollary 1] Both parts of Corollary 1 are labelled (a); the second should be (b).
[Section 6, after Theorem 3] The phrase 'minimises every nondecreasing' should be 'minimises every nonincreasing'.
[Section 7, Lemma 6 proof] The symbol T' is used for both T−{v1,v2} and T−v2; please clarify, and in inequality (6) the sequence should be D_{T−v2}(v1), not D_{T'}(v1) as printed.
[Introduction] In the definition of the hyper-Wiener index, the term d_G(u,) should be d_G(u,v).
[Section 3, Lemma 1] The phrase 'if an only if' should be 'if and only if'.
[Section 2] Please clarify whether 'increasing' is meant strictly; the equality conditions in Proposition 1(b) and Theorem 2(b) depend on strict monotonicity.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main theorems are direct monotonicity transfers from prior distance-sequence lemmas, not fitted predictions.

full rationale

The paper's derivation chain is explicit. Proposition 1 states the monotonicity bridge: if D(G1) <= D(G2), then I(G1) <= I(G2) for any nondecreasing distance-based index I. Theorem 1 is then literally 'a direct consequence' of Lemma 2 and Proposition 1, and Theorem 2 is the same with Lemma 3. Lemmas 2 and 3 are cited from the author's earlier paper [11] rather than proved here, and they are genuinely load-bearing for the claimed Harary-index and hyper-Wiener-index bounds. However, that earlier citation is a prior published extremal result on distance sequences, not a restatement of the target index bounds; it is parameter-free and does not assume any of the Harary/hyper-Wiener inequalities derived here. Thus the dependence is on external support, not a self-referential loop. The newly proved lemmas (Lemma 1, Lemma 5, Lemma 6) are self-contained and independent of the index bounds. No fitted parameter is called a prediction, and no uniqueness or ansatz is smuggled in through self-citation. Minor anomalies such as the apparent inequality reversal in Theorem 2(b) and duplicated labels in Corollary 1 are typos or correctness issues, not circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper's new contributions are pure extremal graph theory. It introduces no free parameters and no new entities. Its load-bearing imports are two extremal distance-sequence lemmas from the author's earlier paper [11] and the k-connectivity of maximal k-degenerate graphs from [7]; the remaining reasoning is standard graph theory.

assumptions (4)

domain assumption For a connected graph of order n and size m, D(G) ≤ D(PK_{n,m}) (Lemma 2 from [11]).
Imported without proof from Casablanca and Dankelmann [11]; it is the entire basis of Theorem 1 and the Harary-index resolution.
domain assumption For even κ and a κ-connected graph of order n, D(G) ≤ D(C_n^{κ/2}) (Lemma 3 from [11]).
Imported from [11]; basis of Theorem 2 for even connectivity.
domain assumption Every maximal k-degenerate graph is k-connected (Lemma 4, cited to Bickle [7]).
Used in Lemma 5 to ensure the removed vertex has degree k and G−v is connected.
standard math Menger's theorem: in a k-connected graph, there are at least k internally disjoint paths from any vertex to any other vertex.
Underlies the claim in Lemma 5 that at least k vertices lie at each distance i from the removed vertex.

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Pith. "Pith review of Distance Sequences to bound the Harary Index and other Wiener-type Indices of a Graph." pith.science (2026). https://pith.science/paper/OZFNDNGB

@misc{pith2026241113439,
  author       = {Pith},
  title        = {Pith review of: Distance Sequences to bound the Harary Index and other Wiener-type Indices of a Graph},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OZFNDNGB}},
  note         = {Machine review of arXiv:2411.13439}
}

abstract

In this paper we obtain bounds on a very general class of distance-based topological indices of graphs, which includes the Wiener index, defined as the sum of the distances between all pairs of vertices of the graph, and most generalisations of the Wiener index, including the Harary index and the hyper-Wiener index. Our results imply several new bounds on well-studied topological indices, among those sharp lower bounds on the Harary index and sharp upper bounds on the hyper-Wiener index for (i) graphs of given order and size (which resolves a problem in the monograph [The Harary index of a graph, Xu, Das, Trinajsti\'{c}, Springer (2015)], (ii) for $\kappa$-connected graphs, where $\kappa$ is even, (iii) for maximal outerplanar graphs and for Apollonian networks (a subclass of maximal planar graphs), and (iv) for trees in which all vertices have odd degree.

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Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [11]

Casablanca, P

R.M. Casablanca, P. Dankelmann, Distance and eccentric seque nces to bound the Wiener index, Hosoya polynomial and the average eccentricity in the strong products of graphs. Discrete Appl. Math. 263 (2019), 105-117

work page 2019

[2] [1]

Alhevaz, M

A. Alhevaz, M. Baghipur, S. Rahimi, Bounds on hyper-Wiener index of graphs. Asian-Eur. J. Math. 10 no. 3 (2017), 1750057, 15 pp

work page 2017

[3] [2]

An and B

X. An and B. Wu, The Wiener index of the kth power of a graph. Appl. Math. Lett. 21 2008, 436-440. 10

work page 2008

[4] [3]

Andriantiana, V

E.O.D. Andriantiana, V. Razanajatovo Misanantenaina, S. Wagne r, Extremal trees with ﬁxed degree sequence Electron. J. Combin. 28 (2021), no. 1, paper 1.1

work page 2021

[5] [4]

G. Ao1, R. Liu, J. Yuan, G. Yu, Wiener-type invariants and k-leaf-connected graphs. Bull. Malays. Math. Sci. Soc. 46 no. 1 (2023), Paper No. 10, 15 pp

work page 2023

[6] [5]

Beineke, R.E

L W. Beineke, R.E. Pippert, The number of labeled k-dimensional trees. J. Combin. Theory 6 (1969), 200-205

work page 1969

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Behtoei, M

A. Behtoei, M. Jannesari, B. Taeri, Maximum Zagreb index, minimum hyper- Wiener index and graph connectivity. Math. Lett. 22 no. 10 (2009), 1571-1576

work page 2009

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Bickle, Structural results on maximal k-degenerate graphs

A. Bickle, Structural results on maximal k-degenerate graphs . Discuss. Math. Graph Theory 32 (2012), 659-676

work page 2012