cp's OEIS Frontend

A180859 Square array read by antidiagonals: T(m,n) is the Wiener index of the windmill graph D(m,n) 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; m>=2, n>=1).

%I A180859 #15 Feb 16 2025 08:33:13
%S A180859 1,3,4,6,14,9,10,30,33,16,15,52,72,60,25,21,80,126,132,95,36,28,114,
%T A180859 195,232,210,138,49,36,154,279,360,370,306,189,64,45,200,378,516,575,
%U A180859 540,420,248,81,55,252,492,700,825,840,742,552,315,100,66,310,621,912,1120,1206,1155,976,702,390,121
%N A180859 Square array read by antidiagonals: T(m,n) is the Wiener index of the windmill graph D(m,n) 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; m>=2, n>=1).
%C A180859 The Wiener index of a connected graph is the sum of the distances between all unordered pairs of nodes in the graph.
%C A180859 For the Wiener indices of D(3,n), D(4,n), D(5,n) and D(6,n) see A033991, A152743, A028994, and A180577, respectively.
%H A180859 B. E. Sagan, Y-N. Yeh and P. Zhang, <a href="http://users.math.msu.edu/users/sagan/Papers/Old/wpg-pub.pdf">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60, 1996, 959-969.
%H A180859 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WindmillGraph.html">Windmill Graph</a>.
%F A180859 T(m,n) = (1/2)n(m-1)((m-1)(2n-1)+1).
%F A180859 The Wiener polynomial of D(m,n) is (1/2)n(m-1)t((m-1)(n-1)t+m).
%e A180859 T(3,2)=14 because the graph D(3,2) consists of two triangles OAB and OCD with a common node O; it has 6 distances equal to 1 (the edges) and 4 distances equal to 2 (AC, AD, BC, and BD); 6 * 1 + 4 * 2 = 14.
%e A180859 Square array starts:
%e A180859    1,   4,   9,  16,  25, ...
%e A180859    3,  14,  33,  60,  95, ...
%e A180859    6,  30,  72, 132, 210, ...
%e A180859   10,  52, 126, 232, 370, ...
%p A180859 T := proc (m, n) options operator, arrow: (1/2)*n*(m-1)*((m-1)*(2*n-1)+1) end proc: for p from 2 to 12 do seq(T(p+1-j, j), j = 1 .. p-1) end do; # yields sequence in triangular form
%o A180859 (PARI) T(m,n) = (1/2)*n*(m-1)*((m-1)*(2*n-1)+1);
%o A180859 antidiag(n) = vector(n-1, k, k; T(n-k+1, k)); \\ _Michel Marcus_, Mar 09 2023
%Y A180859 Cf. A028994, A033991, A152743, A180577.
%K A180859 nonn,tabl
%O A180859 2,2
%A A180859 _Emeric Deutsch_, Sep 25 2010