A192027 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 adjoining m pendant edges at each node of the circuit.

1, 10, 4, 27, 29, 9, 60, 75, 58, 16, 105, 160, 147, 97, 25, 174, 275, 308, 243, 146, 36, 259, 447, 525, 504, 363, 205, 49, 376, 658, 846, 855, 748, 507, 274, 64, 513, 944, 1239, 1371, 1265, 1040, 675, 353, 81, 690, 1278, 1768, 2002, 2022, 1755, 1380, 867, 442, 100

Offset

1,2

Comments

W(1,m) = m^2 = A000290(m).

W(2,m) = A079273(m+1).

W(n,1) = A180574(n).

Links

Table of n, a(n) for n=1..55.

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

If n even, then: W(n,m) = n*(n^2/4 + 2*m^2*n + m^2*n^2/4 + 2*m*n + m*n^2/2 - 2*m)/2;

if n odd, then: W(n,m) = n*(n^2 - 1 + m^2*n^2 + 8*m^2*n - m^2 + 2*m*n^2 + 8*m*n - 10*m)/8.

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(4,9) = 162*t^4 + 360*t^3 + 218*t^2 + 40*t. Its derivative, evaluated at t=1, yields the corresponding Wiener index W(4,9)=4184.

Example

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:
   1,   4,   9,  16,  25,   36,   49, ...;
  10,  29,  58,  97, 146,  205,  274, ...;
  27,  75, 147, 243, 363,  507,  675, ...;
  60, 160, 308, 504, 748, 1040, 1380, ...;

Maple

W := proc (n, m) if `mod`(n, 2) = 0 then (1/2)*n*((1/4)*n^2+2*m^2*n+(1/4)*m^2*n^2+2*m*n+(1/2)*m*n^2-2*m) else (1/8)*(n^2-1+m^2*n^2+8*m^2*n-m^2+2*m*n^2+8*m*n-10*m)*n 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 (1/2)*n*((1/4)*n^2+2*m^2*n+(1/4)*m^2*n^2+2*m*n+(1/2)*m*n^2-2*m) else (1/8)*(n^2-1+m^2*n^2+8*m^2*n-m^2+2*m*n^2+8*m*n-10*m)*n 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(n*(m*t+(1/2)*m*(m-1)*t^2)+n*(sum(t^j, j = 1 .. (1/2)*n-1))*(1+m*t)^2+(1/2)*n*t^((1/2)*n)*(1+m*t)^2)) else sort(expand(n*(m*t+(1/2)*m*(m-1)*t^2)+n*(sum(t^j, j = 1 .. (1/2)*n-1/2))*(1+m*t)^2)) end if end proc: P(4, 9);

Crossrefs

Cf. A000290, A079273, A180574.

Sequence in context: A329649 A040094 A094580 * A180866 A232908 A216653

Adjacent sequences: A192024 A192025 A192026 * A192028 A192029 A192030

Keyword

nonn,tabl

Author

Emeric Deutsch, Jun 26 2011

Status

approved