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%I #14 Mar 07 2017 11:26:38
%S 0,2,10,10,32,32,36,36,88,88,88,106,106,106,222,140,106,284,140,268,
%T 268,222,284,370,536,284,756,330,268,708,222,490,536,268,658,1052,370,
%U 370,708,978,284,872,330,658,1856,756,708,1542,798,1712,658,872,490,2882,1254
%N The total walk count in the rooted tree with Matula-Goebel number n.
%C The total walk count in a graph with n vertices is obtained by counting all walks of lengths 1,2,...,n-1. Some authors define it as 1/2 of the above.
%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.
%C The Maple program yields a(n) by using the command TWC(n).
%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.
%D G. Ruecker and C. Ruecker, Walk counts, labyrinthicity, and complexity of acyclic and cyclic graphs and molecules, J. Chem. Inf. Comput. Sci., 40, 2000, 99-106.
%D G. Ruecker and C. Ruecker, Substructure, subgraph, and walk counts as measures of the complexity of graphs and molecules, J. Chem. Inf. Comput. Sci., 41, 2001, 1457-1462.
%D D. Bonchev and G. A. Buck, Quantitative measures of network complexity, in: Complexity in Chemistry, Biology, and Ecology, Springer, New York, pp. 191-235.
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288.
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F In A193403 it is shown how to find the adjacency matrix of a rooted tree with a given Matula-Goebel number. It is well-known that the (i,j)-entry in the k-th power of the adjacency matrix of a graph G gives the number of walks of length k in G from vertex i to vertex j. The Maple program (improvable) is based on the above facts.
%e a(3)=10 because the rooted tree with Matula-Goebel number 3 is the path a-b-c on 3 vertices and the walks are: ab, ba, bc, cb, abc, cba, aba, bab, bcb, and cbc.
%p with(numtheory); with(linalg): with(LinearAlgebra): V := proc (n) local r, s: 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 = 1 then 1 elif bigomega(n) = 1 then 1+V(pi(n)) else V(r(n))+V(s(n))-1 end if end proc: d := proc (n) local r, s, C, a: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: C := proc (A, B) local c: c := proc (i, j) options operator, arrow: A[1, i]+B[1, j+1] end proc: Matrix(RowDimension(A), RowDimension(B)-1, c) end proc: a := proc (i, j) if i = 1 and j = 1 then 0 elif 2 <= i and 2 <= j then dd[pi(n)][i-1, j-1] elif i = 1 then 1+dd[pi(n)][1, j-1] elif j = 1 then 1+dd[pi(n)][i-1, 1] else end if end proc: if n = 1 then Matrix(1, 1, [0]) elif bigomega(n) = 1 then Matrix(V(n), V(n), a) else Matrix(blockmatrix(2, 2, [dd[r(n)], C(dd[r(n)], dd[s(n)]), Transpose(C(dd[r(n)], dd[s(n)])), SubMatrix(dd[s(n)], 2 .. RowDimension(dd[s(n)]), 2 .. RowDimension(dd[s(n)]))])) end if end proc: for n to 10000 do dd[n] := d(n) end do: DA := proc (d) local aa: aa := proc (i, j) if d[i, j] = 1 then 1 else 0 end if end proc: Matrix(RowDimension(d), RowDimension(d), aa) end proc: TWC := proc (n) options operator, arrow: add(add((sum(DA(d(n))^k, k = 1 .. V(n)-1))[i, j], j = 1 .. V(n)), i = 1 .. V(n)) end proc; seq(TWC(n), n = 1 .. 55);
%Y Cf. A193403.
%K nonn
%O 1,2
%A _Emeric Deutsch_, Feb 20 2012
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