A192028 Square array read by antidiagonals: W(n,m) (n >= 1, m >= 1) is the Wiener index of the graph G(n,m) obtained from the n-circuit graph by joining at each of its nodes a path with m nodes (n >= 1, m >= 1; if m=1, then the n-circuit is not modified).

0, 1, 1, 3, 10, 4, 8, 27, 35, 10, 15, 60, 93, 84, 20, 27, 105, 196, 222, 165, 35, 42, 174, 335, 456, 435, 286, 56, 64, 259, 537, 770, 880, 753, 455, 84, 90, 376, 784, 1212, 1475, 1508, 1197, 680, 120, 125, 513, 1112, 1750, 2295, 2515, 2380, 1788, 969, 165

Offset

1,4

Comments

W(1,m) = A000292(m-1).

W(2,m) = A000447(m) = A000292(2m-2).

W(n,1) = A034828(n).

W(n,2) = A180574(n) (n >= 3).

Links

Table of n, a(n) for n=1..55.

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Formula

W(n,m) = (1/24)*n*m*(3*m*n^2 + 12*n*m^2 - 8*m^2 - 12*n*m + 12*m - 4) if n is even;

W(n,m) = (1/24)*n*m*(3*m*n^2 + 12*n*m^2 - 8*m^2 - 12*n*m + 9*m - 4) if n is odd.

The Wiener polynomial P(n,m;t) of the graph G(n,m) is given in the 3rd Maple program. It gives, for example, P(3,4) = 12*t + 12*t^2 + 12*t^3 + 12*t^4 + 9*t^5 + 6*t^6 + 3*t^7. Its derivative, evaluated at t=1, yields the corresponding Wiener index W(3,4)=222.

Example

a(3,1)=27 because in the graph with vertex set {A,B,C,A',B',C'} and edge set {AB, BC, CA, AA', BB', CC'} we have 6 pairs of vertices at distance 1 (the edges), 6 pairs at distance 2 (A'B, A'C, B'A, B'C, C'A, C'B) and 3 pairs at distance 3 (A'B', B'C', C'A'); 6*1 + 6*2 + 3*3 = 27.
The square array starts:
  0,  1,   4,  10,  20,   35,   56,   84, ...;
  1, 10,  35,  84, 165,  286,  455,  680, ...;
  3, 27,  93, 222, 435,  753, 1197, 1788, ...;
  8, 60, 196, 456, 880, 1508, 2380, 3536, ...;

Maple

W := proc (n, m) if `mod`(n, 2) = 0 then (1/24)*n*m*(3*n^2*m+12*n*m^2-8*m^2-12*n*m+12*m-4) else (1/24)*n*m*(3*n^2*m+12*n*m^2-8*m^2-12*n*m+9*m-4) end if end proc: for n to 10 do seq(W(n-i, i+1), i = 0 .. n-1) end do; # yields the antidiagonals in triangular form
W := proc (n, m) if `mod`(n, 2) = 0 then (1/24)*n*m*(3*n^2*m+12*n*m^2-8*m^2-12*n*m+12*m-4) else (1/24)*n*m*(3*n^2*m+12*n*m^2-8*m^2-12*n*m+9*m-4) end if end proc: for n to 10 do seq(W(n, m), m = 1 .. 10) end do; # yields the first 10 entries of each of rows 1, 2, ..., 10.
P := proc (n, m) if `mod`(n, 2) = 0 then sort(expand(simplify(n*t*(t^m-m*t+m-1)/(1-t)^2+(n*(sum(t^j, j = 1 .. (1/2)*n-1))+(1/2)*n*t^((1/2)*n))*(1-t^m)^2/(1-t)^2))) else sort(expand(simplify(n*t*(t^m-m*t+m-1)/(1-t)^2+n*(sum(t^j, j = 1 .. (1/2)*n-1/2))*(1-t^m)^2/(1-t)^2))) end if end proc: P(3, 4);

Crossrefs

Cf. A000292, A000447, A034828, A180574.

Sequence in context: A131814 A003620 A235923 * A111229 A100984 A045985

Adjacent sequences: A192025 A192026 A192027 * A192029 A192030 A192031

Keyword

nonn,tabl

Author

Emeric Deutsch, Jun 27 2011

Status

approved