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A192029
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).
0
6, 12, 36, 20, 72, 111, 30, 120, 220, 252, 42, 180, 365, 496, 480, 56, 252, 546, 820, 940, 816, 72, 336, 763, 1224, 1550, 1592, 1281, 90, 432, 1016, 1708, 2310, 2620, 2492, 1896, 110, 540, 1305, 2272, 3220, 3900, 4095, 3680, 2682, 132, 660, 1630, 2916, 4280, 5432, 6090, 6040, 5196, 3660
OFFSET
3,1
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.
FORMULA
W(n,1) = A002378(n-1) = n(n-1).
W(n,2) = A049598(n-1).
W(n,m) = (1/6)*n*m*(3*n*m + 3*n*m^2 + 2 - 6*m - 2*m^2) (n >= 3, m >= 1).
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.
EXAMPLE
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.
The square array starts:
6, 36, 111, 252, 480, 816, 1281, ...;
12, 72, 220, 496, 940, 1592, 2492, ...;
20, 120, 365, 820, 1550, 2620, 4095, ...;
30, 180, 546, 1224, 2310, 3900, 6090, ...;
MAPLE
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
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 := 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);
CROSSREFS
Sequence in context: A372702 A196992 A321839 * A166636 A167338 A078402
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Jun 27 2011
STATUS
approved