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A196061
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The multiplicative Wiener index of the rooted tree with Matula-Goebel number n.
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0
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1, 2, 2, 12, 12, 8, 8, 288, 288, 288, 144, 144, 144, 34560, 64, 144, 10368, 64, 13824, 13824, 34560, 10368, 3456, 24883200, 10368, 2985984, 5184, 13824, 4976640, 34560, 1024, 24883200, 13824, 8294400, 746496, 3456, 3456, 4976640, 1327104, 10368, 1492992, 5184, 8294400, 7166361600
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OFFSET
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2,2
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COMMENTS
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The multiplicative Wiener index of a connected graph is the product of the distances between all unordered pairs of vertices in the graph.
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.
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REFERENCES
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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, W. Linert, I. Lukovits, and Z. Tomovic, The multiplicative version of the Wiener index, J. Chem. Inf. Comput. Sci., 40, 2000, 113-116.
I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, On the multiplicative Wiener index and its possible chemical applications, Monatshefte f. Chemie, 131, 2000, 421-427.
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.
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LINKS
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FORMULA
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a(n)=Product(k^c(k), k=1..d), where d is the diameter of the rooted tree with Matula-Goebel number n, and c(k) is the number of pairs of nodes at distance k (all these data are contained in the Wiener polynomial; see A196059). The Maple program is based on the above.
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EXAMPLE
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a(7)=8 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y with distances 1,1,1,2,2,2; product of distances is 8.
a(2^m) = 2^[m(m-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.
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MAPLE
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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: product(k^coeff(W(n), x, k), k = 1 .. degree(W(n))) end proc: seq(a(n), n = 2 .. 45);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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