A180577 The Wiener index of the windmill graph D(6,n). The windmill graph D(m,n) is the graph obtained by taking n copies of the complete graph K_m with a vertex in common (i.e., a bouquet of n pieces of K_m graphs).

15, 80, 195, 360, 575, 840, 1155, 1520, 1935, 2400, 2915, 3480, 4095, 4760, 5475, 6240, 7055, 7920, 8835, 9800, 10815, 11880, 12995, 14160, 15375, 16640, 17955, 19320, 20735, 22200, 23715, 25280, 26895, 28560, 30275, 32040, 33855, 35720, 37635, 39600, 41615, 43680, 45795

Offset: 1

Author

Emeric Deutsch, Sep 21 2010

Keywords

• nonn
• easy

Comments

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

The Wiener polynomial of D(m,n) is (1/2)n(m-1)t[(m-1)(n-1)t+m].

The Wiener index of D(m,n) is (1/2)n(m-1)[(m-1)(2n-1)+1].

For the Wiener indices of D(3,n), D(4,n), and D(5,n) see A033991, A152743, and A028994, respectively.

Links

• B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., Vol. 60, 1996, pp. 959-969.
• Eric Weisstein's World of Mathematics, Windmill Graph.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A028994, A033991, A152743.

Programs

Maple

seq(5*n*(-2+5*n), n = 1 .. 40);

PARI

a(n)=5*n*(5*n-2) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 5*n*(5*n-2).

G.f.: -5*x*(7*x+3)/(x-1)^3. - Colin Barker, Oct 30 2012

From Elmo R. Oliveira, Apr 03 2025: (Start)

E.g.f.: 5*exp(x)*x*(3 + 5*x).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

Extensions

More terms from Elmo R. Oliveira, Apr 03 2025