A192029 Square array read by antidiagonals: W(n,m) (n >= 3, m >= 1) is the Wiener index of the graph G(n,m) obtained from an n-wheel graph by adjoining at each of its n rim nodes a path with m nodes (if m=1, then the n-wheel is not modified).

6, 12, 36, 20, 72, 111, 30, 120, 220, 252, 42, 180, 365, 496, 480, 56, 252, 546, 820, 940, 816, 72, 336, 763, 1224, 1550, 1592, 1281, 90, 432, 1016, 1708, 2310, 2620, 2492, 1896, 110, 540, 1305, 2272, 3220, 3900, 4095, 3680, 2682, 132, 660, 1630, 2916, 4280, 5432, 6090, 6040, 5196, 3660

Offset

3,1

Links

Table of n, a(n) for n=3..57.

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Formula

W(n,1) = A002378(n-1) = n(n-1).

W(n,2) = A049598(n-1).

W(n,m) = (1/6)*n*m*(3*n*m + 3*n*m^2 + 2 - 6*m - 2*m^2) (n >= 3, m >= 1).

The Wiener polynomial P(n,m;t) of the graph G(n,m) is given in the 3rd Maple program. It gives, for example, P(3,4) = 16*t + 18*t^2 + 20*t^3 + 14*t^4 + 8*t^5 + 2*t^6. Its derivative, evaluated at t=1, yields the corresponding Wiener index W(4,3)=220.

Example

W(3,2)=36 because in the graph with vertex set {O,A,B,C,A',B',C'} and edge set {OA, OB, OC, AB, BC, CA, AA', BB', CC'} we have 9 pairs of vertices at distance 1 (the edges), 9 pairs at distance 2 (A'O, A'B, A'C, B'O, B'A, B'C, C'O, C'A, C'B) and 3 pairs at distance 3 (A'B', B'C', C'A'); 9*1 + 9*2 + 3*3 = 36.
The square array starts:
   6,  36, 111,  252,  480,  816, 1281, ...;
  12,  72, 220,  496,  940, 1592, 2492, ...;
  20, 120, 365,  820, 1550, 2620, 4095, ...;
  30, 180, 546, 1224, 2310, 3900, 6090, ...;

Maple

W := proc (n, m) options operator, arrow: (1/6)*n*m*(3*n*m+3*n*m^2+2-6*m-2*m^2) end proc: for n from 3 to 12 do seq(W(n-i, i+1), i = 0 .. n-3) end do; # yields the antidiagonals in triangular form
W := proc (n, m) options operator, arrow: (1/6)*n*m*(3*n*m+3*n*m^2+2-6*m-2*m^2) end proc: for n from 3 to 12 do seq(W(n, m), m = 1 .. 10) end do; # yields the first 10 entries of each of rows 3, 4, ..., 12.
P := proc (n, m) options operator, arrow: sort(expand(simplify(n*t*(t^m-m*t+m-1)/(1-t)^2+n*t*(1-t^m)/(1-t)+n*t*(1-t^m)^2/(1-t)^2+(1/2)*n*(n-3)*t^2*(1-t^m)^2/(1-t)^2))) end proc; P(4, 3);

Crossrefs

Cf. A002378, A049598.

Sequence in context: A268283 A196992 A321839 * A166636 A167338 A078402

Adjacent sequences: A192026 A192027 A192028 * A192030 A192031 A192032

Keyword

nonn,tabl

Author

Emeric Deutsch, Jun 27 2011

Status

approved