A193401 Triangle read by rows: row n contains the coefficients (of the increasing powers of the variable) of the characteristic polynomial of the Laplacian matrix of the rooted tree having Matula-Göbel number n.
0, 1, 0, -2, 1, 0, 3, -4, 1, 0, 3, -4, 1, 0, -4, 10, -6, 1, 0, -4, 10, -6, 1, 0, -4, 9, -6, 1, 0, -4, 9, -6, 1, 0, 5, -20, 21, -8, 1, 0, 5, -20, 21, -8, 1, 0, 5, -20, 21, -8, 1, 0, 5, -18, 20, -8, 1, 0, 5, -18, 20, -8, 1, 0, 5, -18, 20, -8, 1, 0, -6, 35, -56, 36, -10, 1, 0, 5, -16, 18, -8, 1, 0, 5, -18, 20, -8, 1
Offset: 1
Examples
Row 4 is 0, 3, -4, 1 because the rooted tree having Matula-Goebel number 4 is V; the Laplacian matrix is [2,-1,-1; -1,1,0; -1,0,1], having characteristic polynomial x^3 - 4x^2 +3x Triangle starts: 0, 1; 0, -2, 1; 0, 3, -4, 1; 0, 3, -4, 1; 0, -4, 10, -6, 1; 0, -4, 10, -6, 1; 0, -4, 9, -6, 1; 0, -4, 9, -6, 1;
Links
- E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
- F. Göbel, 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 Rev. 10 (1968) 273.
- Index entries for sequences related to Matula-Goebel numbers
Programs
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Maple
with(numtheory): with(linalg): with(LinearAlgebra): 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: AL := proc (a) local ll: ll := proc (i, j) if i = j then add(a[i, k], k = 1 .. RowDimension(a)) else -a[i, j] end if end proc: Matrix(RowDimension(a), RowDimension(a), ll) end proc: 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 1000 do dd[n] := d(n) end do: for n from 1 to 18 do seq(coeff(CharacteristicPolynomial(AL(DA(d(n))), x), x, k), k = 0 .. V(n)) end do; # yields triangle in triangular form
Formula
Let T be a rooted tree with root b. If b has degree 1, then let A be the rooted tree with root c, obtained from T by deleting the edge bc emanating from the root. If b has degree >=2, then A is obtained (not necessarily in a unique way) by joining at b two trees B and C, rooted at b. It is straightforward to express the distance matrix of T in terms of the entries of the distance matrix of A (resp. of B and C). Making use of this, the Maple program (improvable!) finds recursively the distance matrices of the rooted trees with Matula-Göbel numbers 1..1000 (upper limit can be altered), then switches (easily) to the Laplacian matrices and finds the coefficients of their characteristic polynomials.
Comments