|
%I #10 Mar 07 2017 11:22:35
%S 0,1,1,2,2,2,2,3,3,3,3,3,3,4,3,3,4,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
%T 5,4,4,4,5,4,5,4,4,5,4,4,5,5,5,4,5,4,5,5,5,5,4,4,5,5,4,5,5,5,5,4,5,5,
%U 5,5,6,5,5,6,5,5,5,4,6,6,4,4,5,5,5,5,5,5,6,5,5,5,5,5,6,5,5,6,6,5,5,5,5,6,5,5,6,4,6,5,6,5,5,6,5,6,4,5,6,6,5,5,5,6,6,4,6,5,6,5,6,6,5,6,6,5,6,5,6,6,5,6,6,6,5,6,5,5,7,6,6,6,6,6,6
%N The rounded atom-bond connectivity (ABC) of the rooted tree with Matula-Goebel number n (n >= 2).
%C The ABC-index of a graph is defined as the summation of sqrt[(d(u) + d(v) - 2)/d(u)d(v)] over all edges uv of G, where d(w) denotes the degree of the vertex w.
%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.
%D B. Furtula, A. Graovac and D. Vukicevic, Atom-bond connectivity index of trees, Discrete Appl. Math., 157, 2009, 2828-2835.
%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 Y-N. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D I. Gutman, B. Furtula and M. Ivanovic, Notes on trees with minimal atom-bond connectivity index, Comm. Math. Comp. Chem. (MATCH), 67, 2012, 467-482.
%D R. Xing, B. Zhou and Zh. Du, Further results on atom-bond connectivity index of trees, Discrete Appl. Math., 158, 2010, 1536-1545.
%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%D E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Applied Math., 160, 2012, 2314-2322.
%H D. Dimitrov, <a href="http://arxiv.org/abs/1401.0279">On structural properties of trees with minimal atom-bond connectivity index</a>, arXiv1401.0279.
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Rooted tree statistics from Matula numbers</a>, arXiv1111.4288.
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F There are recurrence relations that give the ABC-index of an "elevated" rooted tree (attach a new vertex to the root which becomes the root of the new tree) and of the merge of two rooted trees (identify the two roots). They make use of the sequence of the degrees of the level-1 vertices (denoted by DL in the Maple program).
%F In the Maple program, A(n) gives the actual (not rounded) ABC-index of the rooted tree with Matula number n. For example, A(987654321) = (73/10)sqrt(2) + (7/3)sqrt(6)+(8/5)sqrt(5) + (2/5)sqrt(10); the corresponding tree is the 29-vertex tree given in Fig. 2 of the Deutsch reference.
%e a(5)=2; indeed the rooted tree with Matula-Goebel number 5 is the path PQRS (rooted at P). The edges PQ and RS have endpoints of degrees 1 and 2 and the edge QR has endpoints of degrees 2 and 2; consequently, the contributions of these 3 edges to the ABC-index are 1/sqrt(2), 1/sqrt(2), and 1/sqrt(2), respectively; the ABC-index is 3/sqrt(2) = 2.1213.
%p with(numtheory): A := proc (n) local DL, r, s: DL := proc (n) if n = 2 then [1] elif bigomega(n) = 1 then [1+bigomega(pi(n))] else [op(DL(op(1, factorset(n)))), op(DL(n/op(1, factorset(n))))] end if end proc: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 2 then 0 elif bigomega(n) = 1 then A(pi(n))+sqrt(bigomega(pi(n))/(1+bigomega(pi(n))))+add(sqrt((DL(pi(n))[j]+bigomega(pi(n))-1)/(DL(pi(n))[j]*(1+bigomega(pi(n))))), j = 1 .. nops(DL(pi(n))))-add(sqrt((DL(pi(n))[j]+bigomega(pi(n))-2)/(DL(pi(n))[j]*bigomega(pi(n)))), j = 1 .. nops(DL(pi(n)))) else A(r(n))+A(s(n))-add(sqrt((DL(r(n))[j]+bigomega(r(n))-2)/(DL(r(n))[j]*bigomega(r(n)))), j = 1 .. nops(DL(r(n))))-add(sqrt((DL(s(n))[j]+bigomega(s(n))-2)/(DL(s(n))[j]*bigomega(s(n)))), j = 1 .. nops(DL(s(n))))+add(sqrt((DL(r(n))[j]+bigomega(n)-2)/(DL(r(n))[j]*bigomega(n))), j = 1 .. nops(DL(r(n))))+add(sqrt((DL(s(n))[j]+bigomega(n)-2)/(DL(s(n))[j]*bigomega(n))), j = 1 .. nops(DL(s(n)))) end if end proc: seq(round(A(n)), n = 2 .. 200);
%K nonn
%O 2,4
%A _Emeric Deutsch_, Feb 16 2014
|
