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THE VARIATION OF THE RANDIĆ INDEX WITH REGARD TO MINIMUM AND MAXIMUM DEGREE

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the-variation-of-the-randic-index-with-regard-to-minimum-and-maximumNumber Theorymath.NTmath.RTposed by Milica Milivojević, Ljiljana Pavlovićrecorded: open · 1 machine check, unexamined

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Statement

Conjecture. If G∈G(k,n)G \in G(k,n) and if

p={⌊n2⌋or⌈n2⌉ifn≡1( mod 4),kiseven;n≡3( mod 4),kiseven,⌊n2⌋ifn≡1( mod 4),kisodd,n−22orn+22ifn≡2( mod 4),kiseven,⌈n2⌉ifn≡3( mod 4),kisodd,p=\begin{cases}\left\lfloor\frac{n}{2}\right\rfloor \text{or}\left\lceil\frac{n}{2}\right\rceil&\text{if}n \equiv 1(\bmod 4),k \text{iseven;}n \equiv 3(\bmod 4),k \text{iseven,}\\ \left\lfloor\frac{n}{2}\right\rfloor&\text{if}n \equiv 1(\bmod 4),k \text{isodd,}\\ \frac{n-2}{2}\text{or}\frac{n+2}{2}&\text{if}n \equiv 2(\bmod 4),k \text{iseven,}\\ \left\lceil\frac{n}{2}\right\rceil&\text{if}n \equiv 3(\bmod 4),k \text{isodd,}\end{cases}

then R′(G)≥n2−12(1k−1n−1)p(n−p)ifn2<k≤n−2,R^{\prime}(G)\geq \frac{n}{2}-\frac{1}{2}\left(\frac{1}{k}-\frac{1}{n-1}\right)p(n-p)\quad \text{if}\frac{n}{2}<k \leq n-2, where p and n are given above. Equality holds if and only if G∈Gn,p,kG \in G_{n,p,k} .

Context

Candidate 1 of the open problems stated in "THE VARIATION OF THE RANDIĆ INDEX WITH REGARD TO MINIMUM AND MAXIMUM DEGREE", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
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    NEW

    Problem: For finite simple undirected graphs, reconstruct R′(G)R'(G) as

    R′(G)=∑uv∈E(G)1max⁡{d(u),d(v)},R'(G)=\sum_{uv\in E(G)}\frac1{\max\{d(u),d(v)\}},

    as supported by the paper’s later formula xij/jx_{ij}/j for edges between degrees i≤ji\le j. Let G(k,n)G(k,n) be connected simple nn-vertex graphs with minimum degree exactly kk. Let Gn,p,k{\mathcal G}_{n,p,k} be the complements of graphs consisting of an (n−k−1)(n-k-1)-regular graph on pp vertices and n−pn-p isolated vertices.

    The conjecture asserts the stated lower bound for n/2<k≤n−2n/2<k\le n-2, with equality iff G∈Gn,p,kG\in{\mathcal G}_{n,p,k}.

    Result: The conjecture is false: the equality characterization is wrong.

    Take n=10n=10, k=6k=6. Then n≡2(mod4)n\equiv2\pmod4, kk is even, so the conjecture allows p=4p=4 or p=6p=6, and

    p(10−p)=24,n2−12(1k−1n−1)p(n−p)=5−12(16−19)24=133.p(10-p)=24,\qquad \frac n2-\frac12\Bigl(\frac1k-\frac1{n-1}\Bigr)p(n-p) =5-\frac12\Bigl(\frac16-\frac19\Bigr)24 =\frac{13}{3}.

    Let HH be the graph on vertices 1,…,101,\dots,10 with edges

    16, 46, 1 10, 4 10, 49, 18, 69, 9 10.16,\ 46,\ 1\,10,\ 4\,10,\ 49,\ 18,\ 69,\ 9\,10.

    Let G=H‾G=\overline H. Then

    dH=(3,0,0,3,0,3,0,1,3,3),d_H=(3,0,0,3,0,3,0,1,3,3),

    so

    dG=(6,9,9,6,9,6,9,8,6,6).d_G=(6,9,9,6,9,6,9,8,6,6).

    Thus G∈G(6,10)G\in G(6,10).

    Using

    R′(G)=n2−12∑uv∈E(G)∣1d(u)−1d(v)∣,R'(G)=\frac n2-\frac12\sum_{uv\in E(G)} \left|\frac1{d(u)}-\frac1{d(v)}\right|,

    the nonzero degree-difference contributions are:

    20(16−19)+4(16−18)+4(18−19)=2018+424+472=43.20\Bigl(\frac16-\frac19\Bigr) +4\Bigl(\frac16-\frac18\Bigr) +4\Bigl(\frac18-\frac19\Bigr) =\frac{20}{18}+\frac{4}{24}+\frac{4}{72} =\frac43.

    Hence

    R′(G)=5−12⋅43=133,R'(G)=5-\frac12\cdot\frac43=\frac{13}{3},

    so equality in the conjectured bound holds.

    But G∉G10,4,6∪G10,6,6G\notin{\mathcal G}_{10,4,6}\cup{\mathcal G}_{10,6,6}, because G‾=H\overline G=H has degree sequence

    (3,3,3,3,3,1,0,0,0,0),(3,3,3,3,3,1,0,0,0,0),

    whereas graphs in G10,4,6{\mathcal G}_{10,4,6} or G10,6,6{\mathcal G}_{10,6,6} have complements with degree sequence respectively (34,06)(3^4,0^6) or (36,04)(3^6,0^4).

    So the “if and only if” equality clause is false. A natural repair is to keep the inequality but weaken/replace the equality characterization. In the exceptional case n≡2(mod4)n\equiv2\pmod4, kk even, equality is not unique to Gn,p,k{\mathcal G}_{n,p,k}; the example above is a genuine additional extremal graph.

    Citation: Original conjecture and definitions: Milica Milivojević and Ljiljana Pavlović, “The variation of the Randić index with regard to minimum and maximum degree,” arXiv:1602.03698. The counterexample above is supplied here.

    Reviews

    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification TYPE1

      PASS

      The proposed counterexample is valid. For n=10,k=6n=10,k=6, the allowed pp values are 4,64,6, giving bound 13/313/3. The complement construction has degree sequence in GG equal to (6,9,9,6,9,6,9,8,6,6)(6,9,9,6,9,6,9,8,6,6), so G∈G(6,10)G\in G(6,10). The counted degree-difference contributions give R′(G)=13/3R'(G)=13/3, so equality holds.

      However G‾\overline G has degree sequence (35,1,04)(3^5,1,0^4), not (34,06)(3^4,0^6) or (36,04)(3^6,0^4), so G∉G10,4,6∪G10,6,6G\notin G_{10,4,6}\cup G_{10,6,6}. Thus the “equality iff” part of the conjecture is false. I found no evidence of a prior known identical counterexample in the available search.

      Novelty assessment

      TYPE1

      Classification rationale: Genuinely new if not previously noted, but minor: it is a single small counterexample to the equality characterization only, not to the lower-bound inequality. It would merit at most a corrigendum/short note, not a standalone combinatorics paper.

      Literature check: I found the original arXiv paper and published version, and searched exact-title/DOI/arXiv-ID variants, author-name variants, “counterexample,” “erratum,” “correction,” “equality,” Gn,p,kG_{n,p,k}, G(k,n)G(k,n), R′(G)R'(G), and the specific n=10,k=6n=10,k=6 data in accessible web/index sources. CORE indexed the original article and returned no correction/counterexample record for the DOI; related Randić-index papers found were not this result. I found no prior disproof or stronger correction.

      Citation: Milica Milivojević and Ljiljana Pavlović, “The variation of the Randić index with regard to minimum and maximum degree,” Discrete Applied Mathematics 217 (2017), 286–293, DOI: 10.1016/j.dam.2016.09.010; arXiv:1602.03698.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

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