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
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
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Sep 30 2010
EXTENSIONS
Typos in Wiener polynomial information corrected by Emeric Deutsch, Sep 30 2010
STATUS
approved