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).
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
Examples
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, ...;
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.
Programs
-
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);
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.