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A192026
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Square array read by antidiagonals: W(n,m) (n >= 3, m >= 1) is the Wiener index of the graph G(n,m) obtained from an n-wheel graph by adjoining m pendant edges at each node of the cycle.
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1
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36, 72, 90, 120, 180, 168, 180, 300, 336, 270, 252, 450, 560, 540, 396, 336, 630, 840, 900, 792, 546, 432, 840, 1176, 1350, 1320, 1092, 720, 540, 1080, 1568, 1890, 1980, 1820, 1440, 918, 660, 1350, 2016, 2520, 2772, 2730, 2400, 1836, 1140, 792, 1650, 2520, 3240, 3696, 3822, 3600, 3060, 2280, 1386
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OFFSET
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3,1
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LINKS
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FORMULA
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W(n,m) = n*(n-1)*(m+1)*(2*m+1) (n >= 3, m >= 1).
The Wiener polynomial of the graph G(n,m) is P(n,m;t) = n*(m+2)*t + (1/2)*n*(m^2+n+5*m-3)*t^2 + n*m*(m+n-3)*t^3 + (1/2)*n*m^2*(n-3)*t^4.
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EXAMPLE
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W(3,1)=36 because in the graph with vertex set {O,A,B,C,A',B',C'} and edge set {OA, OB, OC, AB, BC, CA, AA', BB', CC'} we have 9 pairs of vertices at distance 1 (the edges), 9 pairs at distance 2 (A'O, A'B, A'C, B'O, B'A, B'C, C'O, C'A, C'B) and 3 pairs at distance 3 (A'B', B'C', C'A'); 9*1 + 9*2 + 3*3 = 36.
The square array starts:
36, 90, 168, 270, 396, 546, 720, 918, ...;
72, 180, 336, 540, 792, 1092, 1440, 1836, ...;
120, 300, 560, 900, 1320, 1820, 2400, 3060, ...;
180, 450, 840, 1350, 1980, 2730, 3600, 4590, ...;
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MAPLE
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W := proc (n, m) options operator, arrow: n*(n-1)*(m+1)*(2*m+1) end proc: for n from 3 to 12 do seq(W(n-i, i+1), i = 0 .. n-3) end do; # yields the antidiagonals in triangular form
W := proc (n, m) options operator, arrow: n*(n-1)*(m+1)*(2*m+1) end proc: for n from 3 to 12 do seq(W(n, m), m = 1 .. 10) end do; # yields the first 10 entries of each of rows 3, 4, ..., 12.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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