A192031 - OEIS

%I #23 Feb 16 2025 08:33:15

%S 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,

%T 63,27,30,65,80,35,33,77,99,44,36,90,120,54,39,104,143,65,42,119,168,

%U 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

%N 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.

%C The graph G(n) is a special case of the graph G(n,m) defined in A192026 (m=1).

%C T(n,k) is also the number of unordered pairs of nodes at distance k in the gear graph G(n) obtained from a wheel graph with n spokes by adding a node between each pair of adjacent nodes of the cycle (n>=3, k>=1). Example: T(3,3)=3 because in the graph G(3) with vertex set {A,B,C,D,B',C',D'} and edge set {BD', D'C, CB', B'D, DC', C'B,AB,AC,AD} there are exactly 3 pairs of vertices at distance 3: BB', CC', and DD'.

%C Row 3 contains 3 entries; row n>=4 contains 4 entries.

%C Sum of entries in row n is n*(2n+1)=A014105(n).

%C Sum(k*T(n,k),k>=1)=6*n*(n-1)=A049598(n-1) (the Wiener indices).

%H B. E. Sagan, Y-N. Yeh, and P. Zhang, <a href="http://users.math.msu.edu/users/sagan/Papers/Old/wpg-pub.pdf">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60, 1996, 959-969.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WheelGraph.html">Wheel Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HelmGraph.html">Helm Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GearGraph.html">Gear Graph</a>.

%F 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.

%F Conjectures from _Colin Barker_, Mar 28 2019: (Start)

%F 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).

%F 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)

%e 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'.

%e Rectangular array starts:

%e 9,9,3;

%e 12,14,8,2;

%e 15,20,15,5;

%e 18,27,24,9;

%p 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

%t 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 *)

%Y Cf. A014105, A049598, A192026.

%K nonn,tabf,changed

%O 3,1

%A _Emeric Deutsch_, Jun 30 2011