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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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified June 19 16:01 EDT 2021. Contains 345144 sequences. (Running on oeis4.)