A180859 Square array read by antidiagonals: T(m,n) is the Wiener index of the windmill graph D(m,n) obtained by taking n copies of the complete graph K_m with a vertex in common (i.e., a bouquet of n pieces of K_m graphs; m>=2, n>=1).

1, 3, 4, 6, 14, 9, 10, 30, 33, 16, 15, 52, 72, 60, 25, 21, 80, 126, 132, 95, 36, 28, 114, 195, 232, 210, 138, 49, 36, 154, 279, 360, 370, 306, 189, 64, 45, 200, 378, 516, 575, 540, 420, 248, 81, 55, 252, 492, 700, 825, 840, 742, 552, 315, 100, 66, 310, 621, 912, 1120, 1206, 1155, 976, 702, 390, 121

Offset: 2

Author

Emeric Deutsch, Sep 25 2010

Keywords

• nonn
• tabl

Comments

The Wiener index of a connected graph is the sum of the distances between all unordered pairs of nodes in the graph.

For the Wiener indices of D(3,n), D(4,n), D(5,n) and D(6,n) see A033991, A152743, A028994, and A180577, respectively.

Examples

			T(3,2)=14 because the graph D(3,2) consists of two triangles OAB and OCD with a common node O; it has 6 distances equal to 1 (the edges) and 4 distances equal to 2 (AC, AD, BC, and BD); 6 * 1 + 4 * 2 = 14.
Square array starts:
   1,   4,   9,  16,  25, ...
   3,  14,  33,  60,  95, ...
   6,  30,  72, 132, 210, ...
  10,  52, 126, 232, 370, ...
		

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, Windmill Graph.

Crossrefs

Cf. A028994, A033991, A152743, A180577.

Programs

Maple

T := proc (m, n) options operator, arrow: (1/2)*n*(m-1)*((m-1)*(2*n-1)+1) end proc: for p from 2 to 12 do seq(T(p+1-j, j), j = 1 .. p-1) end do; # yields sequence in triangular form

PARI

T(m,n) = (1/2)*n*(m-1)*((m-1)*(2*n-1)+1);
antidiag(n) = vector(n-1, k, k; T(n-k+1, k)); \\ Michel Marcus, Mar 09 2023

Formula

T(m,n) = (1/2)n(m-1)((m-1)(2n-1)+1).

The Wiener polynomial of D(m,n) is (1/2)n(m-1)t((m-1)(n-1)t+m).