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A184156 The Wiener polarity index of the rooted tree with Matula-Goebel number n. 0
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

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.

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.

REFERENCES

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.

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

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

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

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

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

LINKS

Table of n, a(n) for n=1..110.

E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011

Index entries for sequences related to Matula-Goebel numbers

FORMULA

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.

EXAMPLE

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.

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.

MAPLE

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);

CROSSREFS

Cf. A196059

Sequence in context: A109913 A197169 A048052 * A187785 A238277 A258757

Adjacent sequences:  A184153 A184154 A184155 * A184157 A184158 A184159

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Oct 12 2011

STATUS

approved

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Last modified October 19 15:50 EDT 2019. Contains 328223 sequences. (Running on oeis4.)