A192028 Square array read by antidiagonals: W(n,m) (n >= 1, m >= 1) is the Wiener index of the graph G(n,m) obtained from the n-circuit graph by joining at each of its nodes a path with m nodes (n >= 1, m >= 1; if m=1, then the n-circuit is not modified).
0, 1, 1, 3, 10, 4, 8, 27, 35, 10, 15, 60, 93, 84, 20, 27, 105, 196, 222, 165, 35, 42, 174, 335, 456, 435, 286, 56, 64, 259, 537, 770, 880, 753, 455, 84, 90, 376, 784, 1212, 1475, 1508, 1197, 680, 120, 125, 513, 1112, 1750, 2295, 2515, 2380, 1788, 969, 165
Offset: 1
Examples
a(3,1)=27 because in the graph with vertex set {A,B,C,A',B',C'} and edge set {AB, BC, CA, AA', BB', CC'} we have 6 pairs of vertices at distance 1 (the edges), 6 pairs at distance 2 (A'B, A'C, B'A, B'C, C'A, C'B) and 3 pairs at distance 3 (A'B', B'C', C'A'); 6*1 + 6*2 + 3*3 = 27.
The square array starts:
0, 1, 4, 10, 20, 35, 56, 84, ...;
1, 10, 35, 84, 165, 286, 455, 680, ...;
3, 27, 93, 222, 435, 753, 1197, 1788, ...;
8, 60, 196, 456, 880, 1508, 2380, 3536, ...;
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) if `mod`(n, 2) = 0 then (1/24)*n*m*(3*n^2*m+12*n*m^2-8*m^2-12*n*m+12*m-4) else (1/24)*n*m*(3*n^2*m+12*n*m^2-8*m^2-12*n*m+9*m-4) end if end proc: for n to 10 do seq(W(n-i, i+1), i = 0 .. n-1) end do; # yields the antidiagonals in triangular form W := proc (n, m) if `mod`(n, 2) = 0 then (1/24)*n*m*(3*n^2*m+12*n*m^2-8*m^2-12*n*m+12*m-4) else (1/24)*n*m*(3*n^2*m+12*n*m^2-8*m^2-12*n*m+9*m-4) end if end proc: for n to 10 do seq(W(n, m), m = 1 .. 10) end do; # yields the first 10 entries of each of rows 1,2,...,10. P := proc (n, m) if `mod`(n, 2) = 0 then sort(expand(simplify(n*t*(t^m-m*t+m-1)/(1-t)^2+(n*(sum(t^j, j = 1 .. (1/2)*n-1))+(1/2)*n*t^((1/2)*n))*(1-t^m)^2/(1-t)^2))) else sort(expand(simplify(n*t*(t^m-m*t+m-1)/(1-t)^2+n*(sum(t^j, j = 1 .. (1/2)*n-1/2))*(1-t^m)^2/(1-t)^2))) end if end proc: P(3, 4);
Formula
W(n,m) = (1/24)*n*m*(3*m*n^2 + 12*n*m^2 - 8*m^2 - 12*n*m + 12*m - 4) if n is even;
W(n,m) = (1/24)*n*m*(3*m*n^2 + 12*n*m^2 - 8*m^2 - 12*n*m + 9*m - 4) if n is odd.
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) = 12*t + 12*t^2 + 12*t^3 + 12*t^4 + 9*t^5 + 6*t^6 + 3*t^7. Its derivative, evaluated at t=1, yields the corresponding Wiener index W(3,4)=222.
Comments