A180867 Square array read by antidiagonals: T(m,n) is the Wiener index of the Dutch windmill graph D(m,n) (m>=3, n>=1).

3, 8, 14, 15, 40, 33, 27, 78, 96, 60, 42, 144, 189, 176, 95, 64, 228, 351, 348, 280, 138, 90, 352, 558, 648, 555, 408, 189, 125, 500, 864, 1032, 1035, 810, 560, 248, 165, 700, 1230, 1600, 1650, 1512, 1113, 736, 315, 216, 930, 1725, 2280, 2560, 2412, 2079, 1464

Offset

3,1

Comments

The Dutch windmill graph D(m,n) (also called friendship graph) is the graph obtained by taking n copies of the cycle graph C_m with a vertex in common (i.e., a bouquet of n C_m graphs).

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

Links

Table of n, a(n) for n=3..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.

Eric Weisstein's World of Mathematics, Dutch Windmill Graph.

Formula

T(3,n) = A033991(n).

T(4,n) = A014642(n).

T(5,n) = A180579(n).

T(6,n) = A180578(n).

T(m,n) = (1/8)n(m^2-1)(2mn-m-2n+2) if m is odd.

T(m,n) = (1/8)n*m^2*(2mn-m-2n+2) if m is even.

The Wiener polynomial of D(m,n) is n*w(m)+(1/2)n(n-1)r(m)^2, where, denoting S(t,p) = Sum_{j=1..p-1}t^j, we have w(m) = mS(t,m/2) + (1/2)mt^(m/2), r(m) = 2S(t,m/2) + t^(m/2) if m is even and w(m) = mS(t,(m+1)/2), r(m) = 2S(t,(m+1)/2) if m is odd.

Example

Square array starts:
   3,  14,  33,  60,   95, ...
   8,  40,  96, 176,  280, ...
  15,  78, 189, 348,  555, ...
  27, 144, 351, 648, 1035, ...

Maple

T := proc (m, n) if `mod`(m, 2) = 1 then (1/8)*n*(m^2-1)*(2*m*n-m-2*n+2) else (1/8)*n*m^2*(2*m*n-m-2*n+2) end if end proc: for n from 3 to 12 do seq(T(n+1-j, j), j = 1 .. n-2) end do; # yields sequence in triangular form

Crossrefs

Cf. A014642, A033991, A180578, A180579.

Sequence in context: A361363 A067789 A225400 * A343412 A063218 A104656

Adjacent sequences: A180864 A180865 A180866 * A180868 A180869 A180870

Keyword

nonn,tabl

Author

Emeric Deutsch, Sep 30 2010

Extensions

Typos in Wiener polynomial information corrected by Emeric Deutsch, Sep 30 2010

Status

approved