A180865 Square array read by antidiagonals: T(m,n) is the Wiener index of the stacked book graph B(m,n) (m>=1, n>=1). B(m,n) is defined as the graph Cartesian product S(m+1) x P(n), where S(m+1) is the star graph on m+1 nodes and P(n) is the path graph on n nodes. The Wiener index of a connected graph is the sum of distances between all unordered pairs of nodes in the graph.

1, 4, 8, 9, 25, 25, 16, 52, 72, 56, 25, 89, 145, 154, 105, 36, 136, 244, 304, 280, 176, 49, 193, 369, 506, 545, 459, 273, 64, 260, 520, 760, 900, 884, 700, 400, 81, 337, 697, 1066, 1345, 1451, 1337, 1012, 561, 100, 424, 900, 1424, 1880, 2160, 2184, 1920, 1404, 760

Offset: 1

Author

Emeric Deutsch, Sep 28 2010

Keywords

• nonn
• tabl

Comments

T(1,n) = A131423(n).

T(2,n) = A180569(n).

Examples

			T(2,1)=4 because B(2,1) reduces to the path graph P(3) which has 2 pairs of nodes at distance 1 and 1 pair at distance 2.
Square array T(m,n) begins:
1, 8, 25, 56, 105, ...
4, 25, 72, 154, 280, ...
9, 52, 145, 304, 545, ...
16, 89, 244, 506, 900, ...
		

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, Stacked Book Graph.

Crossrefs

Cf. A131423, A180569.

Programs

Maple

T := proc (m, n) options operator, arrow: (1/6)*n*(n^2-(m+1)^2+m*n*(m*n+2*n+6*m)) end proc: for n to 10 do seq(T(n+1-j, j), j = 1 .. n) end do; # yields sequence in triangular form

Formula

T(m,n) = (1/6)n[n^2-(m+1)^2+mn(mn+6m+2n)].

The Wiener polynomial p[n](t) of the graph B(m,n) satisfies the recurrence relation p[n] = p[n-1]+mt+(1/2)m(m-1)t^2+[t+mt+2mt^2+m(m-1)t^3]*sum(t^j,j=0..n-2).