A192030 Square array read by antidiagonals: W(n,p) (n>=1, p>=1) is the Wiener index of the graph G(n,p) obtained in the following way: consider n copies of a star tree with p-1 edges, add a vertex to their union, and connect this vertex with the roots of the star trees.

1, 4, 4, 9, 20, 9, 16, 48, 48, 16, 25, 88, 117, 88, 25, 36, 140, 216, 216, 140, 36, 49, 204, 345, 400, 345, 204, 49, 64, 280, 504, 640, 640, 504, 280, 64, 81, 368, 693, 936, 1025, 936, 693, 368, 81, 100, 468, 912, 1288, 1500, 1500, 1288, 912, 468, 100, 121, 580, 1161, 1696, 2065, 2196, 2065, 1696, 1161, 580, 121

Offset: 1

Author

Emeric Deutsch, Jun 29 2011

Keywords

• nonn
• tabl

Comments

W(n,1)=W(1,n)=n^2=A000290(n).

W(n,2)=W(2,n)=A033579(p)=2*n*(3*n-1).

W(p,n)=W(n,p).

Examples

			W(2,2)=20 because G(2,2) is the path graph with 4 edges; its Wiener index is 4*1+3*2+2*3+1*4=20.
The square array starts:
1,4,9,16,25,36,49,...;
4,20,48,88,140,204,280,...;
9,48,117,216,345,504,693,...;
16,88,216,400,640,936,1288,...;
		

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.
• Stephan Wagner, A class of trees and its Wiener index, Acta Applic. Mathem. 91 (2) (2006) 119-132.

Crossrefs

Cf. A000290, A033579

Programs

Maple

W := proc (n, p) options operator, arrow; n*p*(2*n*p-n-p+1) end proc: for n to 11 do seq(W(n-i, i+1), i = 0 .. n-1) end do; # yields sequence in triangular form
W := proc (n, p) options operator, arrow; n*p*(2*n*p-n-p+1) end proc: for n to 7 do seq(W(n, p), p = 1 .. 10) end do; # yields the first 10 entries in each of the first 7 rows

Formula

W(n,p)=n*p*(2*n*p-n-p+1).

The Wiener polynomial of the graph G(n,p) is a*t+b*t^2+c*t^3+d*t^4, where a=n*p, b=(1/2)*n*(n+p^2-p-1), c=n*(n-1)*(p-1), d=(1/2)*n*(n-1)*(p-1)^2.