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%I #11 Apr 02 2017 03:22:03
%S 6,12,36,20,72,111,30,120,220,252,42,180,365,496,480,56,252,546,820,
%T 940,816,72,336,763,1224,1550,1592,1281,90,432,1016,1708,2310,2620,
%U 2492,1896,110,540,1305,2272,3220,3900,4095,3680,2682,132,660,1630,2916,4280,5432,6090,6040,5196,3660
%N Square array read by antidiagonals: W(n,m) (n >= 3, m >= 1) is the Wiener index of the graph G(n,m) obtained from an n-wheel graph by adjoining at each of its n rim nodes a path with m nodes (if m=1, then the n-wheel is not modified).
%H B. E. Sagan, Y-N. Yeh and P. Zhang, <a href="http://dx.doi.org/10.1002/(SICI)1097-461X(1996)60:5<959::AID-QUA2>3.0.CO;2-W">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60, 1996, 959-969.
%F W(n,1) = A002378(n-1) = n(n-1).
%F W(n,2) = A049598(n-1).
%F W(n,m) = (1/6)*n*m*(3*n*m + 3*n*m^2 + 2 - 6*m - 2*m^2) (n >= 3, m >= 1).
%F 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) = 16*t + 18*t^2 + 20*t^3 + 14*t^4 + 8*t^5 + 2*t^6. Its derivative, evaluated at t=1, yields the corresponding Wiener index W(4,3)=220.
%e W(3,2)=36 because in the graph with vertex set {O,A,B,C,A',B',C'} and edge set {OA, OB, OC, AB, BC, CA, AA', BB', CC'} we have 9 pairs of vertices at distance 1 (the edges), 9 pairs at distance 2 (A'O, A'B, A'C, B'O, B'A, B'C, C'O, C'A, C'B) and 3 pairs at distance 3 (A'B', B'C', C'A'); 9*1 + 9*2 + 3*3 = 36.
%e The square array starts:
%e 6, 36, 111, 252, 480, 816, 1281, ...;
%e 12, 72, 220, 496, 940, 1592, 2492, ...;
%e 20, 120, 365, 820, 1550, 2620, 4095, ...;
%e 30, 180, 546, 1224, 2310, 3900, 6090, ...;
%p W := proc (n, m) options operator, arrow: (1/6)*n*m*(3*n*m+3*n*m^2+2-6*m-2*m^2) end proc: for n from 3 to 12 do seq(W(n-i, i+1), i = 0 .. n-3) end do; # yields the antidiagonals in triangular form
%p W := proc (n, m) options operator, arrow: (1/6)*n*m*(3*n*m+3*n*m^2+2-6*m-2*m^2) end proc: for n from 3 to 12 do seq(W(n, m), m = 1 .. 10) end do; # yields the first 10 entries of each of rows 3,4,...,12.
%p P := proc (n, m) options operator, arrow: sort(expand(simplify(n*t*(t^m-m*t+m-1)/(1-t)^2+n*t*(1-t^m)/(1-t)+n*t*(1-t^m)^2/(1-t)^2+(1/2)*n*(n-3)*t^2*(1-t^m)^2/(1-t)^2))) end proc; P(4, 3);
%Y Cf. A002378, A049598.
%K nonn,tabl
%O 3,1
%A _Emeric Deutsch_, Jun 27 2011