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 A184156 The Wiener polarity index of the rooted tree with Matula-Goebel number n. 0

%I

%S 0,0,0,0,1,1,0,0,2,2,2,2,2,2,3,0,2,4,0,3,3,3,4,3,4,4,6,4,3,5,3,0,4,3,

%T 4,6,3,3,5,4,4,6,4,4,7,6,5,4,4,6,4,6,0,9,5,6,4,5,3,7,6,4,8,0,6,6,3,4,

%U 7,7,4,8,6,6,8,6,5,8,4,5,12,5,6,9,5,6,6,5,4,10,6,8,5,7,5,5,6,8,8,8,6,6,9,8,9,4,6,12,5,7

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

%C The Wiener polarity index of a connected graph G is the number of unordered pairs {i,j} of vertices of G such that the distance between i and j is 3.

%C The Matula-Goebel number of a rooted tree is 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.

%D H. Deng, H. Xiao and F. Tang, On the extremal Wiener polarity index of trees with a given diameter, MATCH, Commun. Math. Comput. Chem., 63, 2010, 257-264.

%D W. Du, X. Li and Y. Shi, Algorithms and extremal problem on Wiener polarity index, MATCH, Commun. Math. Comput. Chem., 62, 2009, 235-244.

%D F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.

%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.

%D I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.

%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.

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

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

%F a(n) is the coefficient of x^3 in the Wiener polynomial of the rooted tree with Matula-Goebel number n. The coefficients of these Wiener polynomials are given in A196059. The Maple program is based on the above.

%e a(7)=0 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y with no pair of vertices at distance 3.

%e a(11) = 2 because the rooted tree with Matula-Goebel number 7 is a path on 5 vertices, say a, b, c, d, e, with each of the pairs {a,d} and {b,e} at distance 3.

%p with(numtheory): WP := 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(WP(pi(n))+x*R(pi(n))+x)) else sort(expand(WP(r(n))+WP(s(n))+R(r(n))*R(s(n)))) end if end proc: a := proc (n) options operator, arrow: coeff(WP(n), x, 3) end proc: seq(a(n), n = 1 .. 110);

%Y Cf. A196059

%K nonn

%O 1,9

%A _Emeric Deutsch_, Oct 12 2011

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Last modified August 9 18:32 EDT 2020. Contains 336326 sequences. (Running on oeis4.)