

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 medge star with the center of an nedge star. The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.


0



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



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(ni, 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


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AUTHOR



STATUS

approved



