A180855 Square array read by antidiagonals: T(m,n) is the Wiener index of the banana tree B(n,k) (n>=1, k>=2). B(n,k) is the graph obtained by taking n copies of a star graph on k nodes and connecting with an edge one leaf of each of these n stars with an additional node.

4, 20, 10, 48, 56, 18, 88, 138, 108, 28, 140, 256, 270, 176, 40, 204, 410, 504, 444, 260, 54, 280, 600, 810, 832, 660, 360, 70, 368, 826, 1188, 1340, 1240, 918, 476, 88, 468, 1088, 1638, 1968, 2000, 1728, 1218, 608, 108, 580, 1386, 2160, 2716, 2940, 2790, 2296, 1560, 756, 130

Offset: 1

Author

Emeric Deutsch, Sep 24 2010

Keywords

• nonn
• tabl

Comments

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

Examples

			T(1,2)=4 because the banana tree B(1,2) reduces to a path on 3 nodes, where the distances are 1, 1, and 2.
Square array T(n,k) begins:
4,10,18,28,40,54,70;
20,56,108,176,260,360,476;
48,138,270,444,660,918,1218;
88,256,504,832,1240,1728,2296;
		

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, Banana Tree.

Crossrefs

Cf. A033579, A060787

Programs

Maple

T := proc (n, k) options operator, arrow: n*(k-1)*(3*n*k-2*k+2) end proc: for n to 10 do seq(T(n+2-j, j), j = 2 .. n+1) end do; # yields sequence in triangular form

Mathematica

T[n_, k_] := n*(k - 1)*(3*n*k - 2*k + 2);
Table[T[n - k + 2, k], {n, 1, 10}, {k, 2, n + 1}] // Flatten (* Jean-François Alcover, Aug 26 2024 *)

Formula

T(n,k) = n(k-1)(3nk-2k+2).

T(n,2) = A033579(n).

T(n,4) = A060787(n+2).

The Wiener polynomial of the tree B(n,k) is W(n,k,t)=(1/2)nt(a+bt+ct^2+dt^3+et^4+ft^5), where a=2k, b=3+n+k^2-3k, c=2n+2k-6, d=(n-1)(2k-3), e=2(n-1)(k-2), and f=(n-1)(k-2)^2.