A228309 The hyper-Wiener index of the odd graph O_n (n>=2).

3, 105, 2590, 57015, 1165626, 22834812, 433178460, 8036703675, 146451924190, 2632740298188, 46790614294788, 824017920352900, 14397367664647800, 249906966022292400, 4312825574857068600, 74063143648813911075

Offset: 2

Author

Emeric Deutsch, Aug 20 2013

Keywords

• nonn

Comments

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.

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.

Crossrefs

Cf. A136328, A228308

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: HWi := proc (n) options operator, arrow: subs(t = 1, diff(H(n), t)+(1/2)*(diff(H(n), `$`(t, 2)))) end proc: seq(HWi(n), n = 2 .. 20);

Formula

A formula is "hidden" in the Maple program. B(n) and C(n) are the intersection arrays of O_n, H(n) is the Hosoya-Wiener polynomial of O_n, and HWi(n) is the hyper-Wiener index of O_n.