A228308 Triangle read by rows: T(n,k) (n>=2, 1<=k<=n-1) is the number of unordered pairs of vertices at distances k in the odd graph O_n.
3, 15, 30, 70, 210, 315, 315, 1260, 2520, 3780, 1386, 6930, 17325, 34650, 46200, 6006, 36036, 108108, 270270, 450450, 600600, 25740, 180180, 630630, 1891890, 3783780, 6306300, 7882875, 109395, 875160, 3500640, 12252240, 28588560, 57177120
Offset: 2
Examples
Row 2 has only one entry equal to 3; indeed, O_2 is the complete graph K_3, having 3 distances equal to 1.
References
- N. Biggs, Algebraic Graph Theory, Cambridge, 2nd. Ed., 1993, p. 161.
- R. Balakrishnan, N. Sridharan and K. Viswanathan Iyer, The Wiener index of odd graphs, J. Indian. Inst. Sci., vol. 86, 2006, 527-531.
Links
- R. Balakrishnan, N. Sridharan and K. Viswanathan Iyer,, The Wiener index of odd graphs, J. Ind. Inst. Sci., vol. 86, no. 5, 2006. [Cached copy]
- Eric Weisstein's World of Mathematics, Odd Graph.
Programs
-
Maple
B := proc (n) options operator, arrow: [seq(n-floor((1/2)*m), m = 1 .. n-1)] end proc: C := proc (n) options operator, arrow: [seq(ceil((1/2)*m), m = 1 .. n-1)] end proc: H := proc (n) options operator, arrow: (1/2)*binomial(2*n-1, n-1)*(sum((product(B(n)[r]/C(n)[r], r = 1 .. j))*t^j, j = 1 .. n-1)) end proc: for n from 2 to 10 do seq(coeff(H(n), t, k), k = 1 .. n-1) end do; # yields sequence in triangular form
Formula
A formula is "hidden" in the Maple program. B(n) and C(n) are the intersection arrays of O_n while H(n) is the Hosoya-Wiener polynomial of O_n.
Comments