A171437 Square array read by antidiagonals: T(m,n) is the Wiener index of the prism graph C(m) x P(n) (m>=3, n>=1). The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.
3, 8, 21, 15, 48, 63, 27, 85, 136, 138, 42, 144, 235, 288, 255, 64, 217, 387, 490, 520, 423, 90, 320, 574, 792, 875, 848, 651, 125, 441, 832, 1162, 1395, 1415, 1288, 948, 165, 600, 1134, 1664, 2030, 2232, 2135, 1856, 1323, 216, 781, 1525, 2250, 2880, 3227
Offset: 3
Examples
Square array starts: 3, 21, 63, 138, 255,... 8, 48, 136, 288, 520,... 15, 85, 235, 490, 875,... 27, 144, 387, 792, 1395,...
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, Prism Graph.
Crossrefs
Cf. A138179.
Programs
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Maple
T := proc (m, n) if `mod`(m, 2) = 1 then (1/24)*m*n*(4*m*(n^2-1)+3*n*(m^2-1)) else (1/24)*m^2*n*(4*n^2-4+3*m*n) end if end proc: for m from 3 to 12 do seq(T(m+1-j, j), j = 1 .. m-2) end do; # yields sequence in triangular form
Formula
T(m,n) = (1/24)*m*n*(4*m*(n^2-1)+3*n*(m^2-1)) if m is odd.
T(m,n) = (1/24)*m^2*n*(4*(n^2-1)+3*m*n) if m is even.
The Wiener polynomial p[n](t) for C(2m+1) x P(n) satisfies the recurrence relation p[n]=p[n-1]+p[1]+(2m+1)*a[n], where a[n]=[t+2*sum(t^j,j=2..m+1)*sum(t^j,j=0..n-2) and p[1]=(2m+1)*sum(t^j,j=1..m).
The Wiener polynomial q[n](t) for C(2m) x P(n) satisfies the recurrence relation q[n]=q[n-1]+q[1]+2m*b[n], where b[n]=[t+t^{m+1}+2*sum(t^j,j=2..m)*sum(t^j,j=0..n-2) and q[1]=2m*sum(t^j,j=1..m-1)+m*t^m.
The above recurrence relations yield simple expressions for the Wiener polynomials (however, they are too long for being reproduced here).
Comments