A228310 The hyper-Wiener index of the hypercube graph Q(n) (n>=2).

10, 72, 448, 2560, 13824, 71680, 360448, 1769472, 8519680, 40370176, 188743680, 872415232, 3992977408, 18119393280, 81604378624, 365072220160, 1623497637888, 7181185318912, 31610959298560, 138538465099776, 604731395276800, 2630031813640192

Offset: 2

Author

Emeric Deutsch, Aug 20 2013

Keywords

• nonn
• easy

Comments

The hypercube graph Q(n) has as vertices the binary words of length n and an edge joins two vertices whenever the corresponding binary words differ in just one place.

Q(n) is distance-transitive and therefore also distance-regular. The intersection array is {n,n-1,n-2,...,1; 1,2,3,...,n-1,n}.

References

• Norman Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993 (p. 161).

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, Hypercube Graph.
• Index entries for linear recurrences with constant coefficients, signature (12,-48,64)

Crossrefs

Cf. A143376, A002697

Programs

Maple

a := proc (n) options operator, arrow: 4^(n-2)*n*(3+n) end proc: seq(a(n), n = 2 .. 25);

Mathematica

LinearRecurrence[{12,-48,64},{10,72,448},30] (* Harvey P. Dale, Dec 13 2024 *)

Formula

a(n) = 4^{n-2}*n*(3+n).

G.f.: 2*x^2*(5 - 24*x + 32*x^2)/(1-4*x)^3.

The Hosoya-Wiener polynomial of Q(n) is 2^{n-1}*((1+t)^n - 1).