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The hyper-Wiener index of the rooted tree with Matula-Goebel number n.
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%I #24 Jun 21 2024 03:48:04

%S 0,1,5,5,15,15,12,12,35,35,35,28,28,28,70,22,28,54,22,58,58,70,54,44,

%T 126,54,90,47,58,99,70,35,126,58,108,76,44,44,99,84,54,83,47,108,150,

%U 90,99,63,91,165,108,83,35,118,210,69,84,99,58,131,76,126,129,51,170,170,44,91,150,143,84,101,83,76,231

%N The hyper-Wiener index of the rooted tree with Matula-Goebel number n.

%C The hyper-Wiener index of a connected graph is (1/2)*Sum [d(i,j)+d(i,j)^2], where d(i,j) is the distance between the vertices i and j and summation is over all unordered pairs of vertices (i,j).

%C The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m >= 2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.

%H G. G. Cash, <a href="http://dx.doi.org/10.1016/S0893-9659(02)00059-9">Relationship between the Hosoya polynomial and the hyper-Wiener index</a>, Appl. Math. Letters, 15, 2002, 893-895.

%H Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288 [math.CO], 2011.

%H F. Goebel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.

%H I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.

%H I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.

%H D. J. Klein, I. Lukovits and I. Gutman, <a href="https://doi.org/10.1021/ci00023a007">On the definition of the hyper-Wiener index for cycle-containing structures</a>, J. Chem. Inf. Comput. Sci., 35, 1995, 50-52.

%H D. W. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.

%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>

%F a(n) = W'(n,1) + (1/2)W"(n,1), where W(n,x) is the Wiener polynomial (also called Hosoya polynomial) of the rooted tree with Matula-Goebel index n. W(n)=W(n,x) is obtained recursively in A196059. The Maple program is based on the above.

%e a(7)=12 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y; the distances are 1,1,1,2,2,2; sum of distances = 9; sum of squared distances = 15; (9+15)/2=12.

%e a(2^m) = m(3m-1)/2 because the rooted tree with Matula-Goebel number 2^m is a star with m edges and we have m distances 1 and m(m-1)/2 distances 2; sum of the distances = m^2; sum of the squared distances = 2m^2 - m; hyper-Wiener index is (1/2)(3m^2 - m).

%p with(numtheory): W := proc (n) local r, s, R: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: R := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(x*R(pi(n))+x)) else sort(expand(R(r(n))+R(s(n)))) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(W(pi(n))+x*R(pi(n))+x)) else sort(expand(W(r(n))+W(s(n))+R(r(n))*R(s(n)))) end if end proc: a := proc (n) options operator, arrow: subs(x = 1, diff(W(n), x)+(1/2)*(diff(W(n), `$`(x, 2)))) end proc: seq(a(n), n = 1 .. 75);

%t r[n_] := FactorInteger[n][[1, 1]];

%t s[n_] := n/r[n];

%t R[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, x*R[PrimePi[n]] + x, True, R[r[n]] + R[s[n]]];

%t W[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, W[PrimePi[n]] + x*R[PrimePi[n]] + x, True, W[r[n]] + W[s[n]] + R[r[n]]*R[s[n]]];

%t a[n_] := (D[W[n], x] /. x -> 1) + (1/2)*(D[W[n], {x, 2}] /. x -> 1);

%t Table[a[n], {n, 1, 75}] (* _Jean-François Alcover_, Jun 19 2024, after Maple code *)

%Y Cf. A196059.

%K nonn

%O 1,3

%A _Emeric Deutsch_, Sep 30 2011