A192031 Rectangular array read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the helm graph G(n) obtained from a wheel graph with n spokes by adjoining a pendant edge at each node of the cycle (n>=3, k>=1). The entries in row n are the coefficients of the corresponding Wiener polynomial.
9, 9, 3, 12, 14, 8, 2, 15, 20, 15, 5, 18, 27, 24, 9, 21, 35, 35, 14, 24, 44, 48, 20, 27, 54, 63, 27, 30, 65, 80, 35, 33, 77, 99, 44, 36, 90, 120, 54, 39, 104, 143, 65, 42, 119, 168, 77, 45, 135, 195, 90, 48, 152, 224, 104, 51, 170, 255, 119, 54, 189, 288, 135, 57, 209, 323, 152, 60, 230, 360, 170
Offset: 3
Examples
T(3,3)=3 because in the graph G(3) with vertex set {A,B,C,D,B',C',D'} and edge set {BC,CD,DB,AB,AC,AD,BB',CC",DD'} there are exactly 3 pairs of vertices at distance 3: B'C', C'D', and D'B'.
Rectangular array starts:
9,9,3;
12,14,8,2;
15,20,15,5;
18,27,24,9;
Links
- B. E. Sagan, Y-N. Yeh, and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
- Eric Weisstein's World of Mathematics, Wheel Graph.
- Eric Weisstein's World of Mathematics, Helm Graph.
- Eric Weisstein's World of Mathematics, Gear Graph.
Programs
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Maple
P := proc (n) options operator, arrow: 3*n*t+(1/2)*n*(n+3)*t^2+n*(n-2)*t^3+(1/2)*n*(n-3)*t^4 end proc: T := proc (n, k) options operator, arrow: coeff(P(n), t, k) end proc: seq(T(3, k), k = 1 .. 3); for n from 4 to 20 do seq(T(n, k), k = 1 .. 4) end do; # yields rows 3,4,..., 20 of the rectangular array
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Mathematica
P[n_] := 3*n*t + (1/2)*n*(n+3)*t^2 + n*(n-2)*t^3 + (1/2)*n*(n-3)*t^4; T[n_]:=Rest@CoefficientList[P[n], t]; Table[T[n], {n, 3, 20}] // Flatten (* Jean-François Alcover, Sep 07 2024, after Maple program *)
Formula
Generating polynomial of row n (i.e. the Wiener polynomial of the graph G(n)) is P(n;t)=3*n*t+(1/2)*n*(n+3)*t^2+n*(n-2)*t^3+(1/2)*n*(n-3)*t^4.
Conjectures from Colin Barker, Mar 28 2019: (Start)
G.f.: x^3*(9 + 3*x^2 + 9*x^3 - 22*x^4 + 3*x^5 - 10*x^6 - 11*x^7 + 16*x^8 + 2*x^9 + 6*x^10 + 3*x^11 - 3*x^12 - 3*x^13) / ((1 - x)^3*(1 + x)^2*(1 + x^2)^3).
a(n) = a(n-1) - a(n-2) + a(n-3) + 2*a(n-4) - 2*a(n-5) + 2*a(n-6) - 2*a(n-7) - a(n-8) + a(n-9) - a(n-10) + a(n-11) for n>16. (End)
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