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

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Last modified August 15 07:32 EDT 2024. Contains 375173 sequences. (Running on oeis4.)