OFFSET
2,1
COMMENTS
Row n contains n-1 entries (n>=2).
The odd graph O_n is a graph whose vertices represent the (n-1)-subsets of {1,2,...,2n-1} and two vertices are connected if and only if they correspond to disjoint subsets. It is a distance regular graph.
The entries in row n are the coefficients of the Hosoya-Wiener polynomial of the odd graph O_n (n>=2).
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.
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.
EXAMPLE
Row 2 has only one entry equal to 3; indeed, O_2 is the complete graph K_3, having 3 distances equal to 1.
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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Aug 20 2013
STATUS
approved