A192032 - OEIS

A192032

Square array read by antidiagonals: W(m,n) (m >= 0, n >= 0) is the Wiener index of the graph G(m,n) obtained in the following way: connect by an edge the center of an m-edge star with the center of an n-edge star. The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.

1, 4, 4, 9, 10, 9, 16, 18, 18, 16, 25, 28, 29, 28, 25, 36, 40, 42, 42, 40, 36, 49, 54, 57, 58, 57, 54, 49, 64, 70, 74, 76, 76, 74, 70, 64, 81, 88, 93, 96, 97, 96, 93, 88, 81, 100, 108, 114, 118, 120, 120, 118, 114, 108, 100, 121, 130, 137, 142, 145, 146, 145, 142, 137, 130, 121

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OFFSET

0,2

COMMENTS

W(n,0) = W(0,n) = A000290(n+1) = (n+1)^2.

W(n,1) = W(1,n) = A028552(n+1) = (n+1)*(n+4).

W(n,2) = W(2,n) = A028881(n+4) = n^2 + 8*n + 9.

W(n,n) = A079273(n+1) = 5*n^2 + 4*n + 1.

W(n,m) = W(m,n) (trivially).

LINKS

Table of n, a(n) for n=0..65.

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(m,n) = m^2 + n^2 + 3*m*n + 2*m + 2*n + 1.

The Wiener polynomial of the graph G(n,m) is P(m,n;t) = (m+n+1)*t + (1/2)*(m^2 + n^2 + m + n)*t^2 + m*n*t^3.

EXAMPLE

W(1,2)=18 because in the graph with vertex set {A,a,B,b,b'} and edge set {AB, Aa, Bb, Bb'} we have 4 pairs of vertices at distance 1 (the edges), 4 pairs at distance 2 (Ab, Ab', Ba, bb') and 2 pairs at distance 3 (ab,ab'); 4*1 + 4*2 + 2*3 = 18.

The square array starts:

1, 4, 9, 16, 25, ...;

4, 10, 18, 28, 30, ...;

9, 18, 29, 42, 57, ...;

16, 28, 42, 58, 76, ...;

MAPLE

W := proc (m, n) options operator, arrow: m^2+n^2+3*m*n+2*m+2*n+1 end proc: for n from 0 to 10 do seq(W(n-i, i), i = 0 .. n) end do; # yields the antidiagonals in triangular form

W := proc (m, n) options operator, arrow: m^2+n^2+3*m*n+2*m+2*n+1 end proc: for m from 0 to 9 do seq(W(m, n), n = 0 .. 9) end do; # yields the first 10 entries of each of rows 0, 1, 2, ..., 9

CROSSREFS

Cf. A000290, A028552, A028881, A079273.

Sequence in context: A075709 A332777 A238629 * A116682 A168157 A222045

Adjacent sequences: A192029 A192030 A192031 * A192033 A192034 A192035

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Jun 30 2011

STATUS

approved