bibkey: kimberling2012a192023 authors: Clark Kimberling year: 2012 title: “OEIS A192023, The Wiener index of the comb-shaped graph |||…|| with 2n (n>=1) nodes” doi: null url: https://oeis.org/A192023 claim: “The Wiener index of the comb-shaped graph |||…|| with 2n (n>=1) nodes. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph. Conjecture: for n>2, A192023(n-2) is the number of 2 X 2 matrices with all terms in {1,2,…,n} and determinant 2n. - Clark Kimberling, Mar 31 2012” strata_touched:

• D5/S0/Certificates/KimberlingCombWienerDeterminantRefutation license: citation-only triage: anchor

OEIS A192023

The NAME of A192023 defines the graph statistic:

The Wiener index of the comb-shaped graph |||…|_| with 2n (n>=1) nodes. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph.

The FORMULA line gives a(n) = n*(2*n^2 + 6*n - 5)/3. The 2022 paper Wiener Index of Some Brooms proves this graph-index formula, not the matrix-count comment. The literal comment fails already at n = 3: the two-vertex comb has index one and exactly two allowed matrices have determinant six. No corrected comment or exhaustive literature claim follows.

Verified locator

• URL: https://oeis.org/A192023
• NAME (verbatim): The Wiener index of the comb-shaped graph |||…|_| with 2n (n>=1) nodes. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph.
• FORMULA (verbatim): a(n) = n*(2n^2 + 6n - 5)/3.
• COMMENTS conjecture line (verbatim): Conjecture: for n>2, A192023(n-2) is the number of 2 X 2 matrices with all terms in {1,2,…,n} and determinant 2n. - Clark Kimberling, Mar 31 2012