OFFSET
1,2
COMMENTS
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.
The eccentric connectivity index of a simple connected graph G is defined as the sum over all vertices i of G of the product E(i) D(i), where E(i) is the eccentricity and D(i) is the degree of vertex i.
REFERENCES
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 Y-N. 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.
V. Sharma, R. Goswami, and A. K. Madan, Eccentric Connectivity index: a novel highly discriminating topological descriptor for structure-property and structure-activity studies, J. Chem. Inf. Comput. Sci., 37, 1997, 273-282.
LINKS
FORMULA
Explanation of the Maple program: "V" finds recursively the number of vertices (needed later); "d" finds recursively the distance matrix; "a" finds the adjacency matrix from the distance matrix; "RS" finds the vector of the row sums of any matrix (will be applied to the adjacency matrix to yield the vertex degrees); "MX" finds the vector of the largest row entries of any matrix (will be applied to the distance matrix to yield the vertex eccentricities); "ECI" finds the eccentric connectivity index by taking the dot product of the two vectors just mentioned.
EXAMPLE
a(7)=9 because the rooted tree with Matula-Goebel number 7 is Y; the 3 pendant vertices have degree 1 and eccentricity 2 and the 4th vertex has degree 3 and eccentricity 1; 1*2 + 1*2 + 1*2 + 3*1 = 9.
MAPLE
with(numtheory): 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, dt, drs: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: dt := proc (i, j) if i = 1 and j = 1 then 0 elif i = 1 and 1 < j then 1+dd[pi(n)][1, j-1] elif 1 < i and j = 1 then 1+dd[pi(n)][i-1, 1] elif 1 < i and 1 < j then dd[pi(n)][i-1, j-1] else end if end proc: drs := proc (i, j) if 1 <= i and 1 <= j and i <= V(r(n)) and j <= V(r(n)) then dd[r(n)][i, j] elif 1+V(r(n)) <= i and 1+V(r(n)) <= j and i <= V(n) and j <= V(n) then dd[s(n)][i-V(r(n))+1, j-V(r(n))+1] elif 1 <= i and i <= V(r(n)) and 1+V(r(n)) <= j and j <= n then dd[r(n)][i, 1]+dd[s(n)][1, j-V(r(n))+1] else dd[r(n)][1, j]+dd[s(n)][i-V(r(n))+1, 1] end if end proc: if n = 1 then Matrix(1, 1, [0]) elif bigomega(n) = 1 then Matrix(V(n), V(n), dt) else Matrix(V(n), V(n), drs) end if end proc: for n to 1000 do dd[n] := d(n) end do: > a := proc (n) local ddd, aa: ddd := proc (n) options operator, arrow: d(n) end proc: aa := proc (i, j) if ddd(n)[i, j] = 1 then 1 else 0 end if end proc: Matrix(RowDimension(ddd(n)), RowDimension(ddd(n)), aa) end proc: > RS := proc (m) local dim: dim := RowDimension(m): Matrix(1, dim, [seq(add(m[i, j], j = 1 .. dim), i = 1 .. dim)]) end proc: > MX := proc (m) local dim: dim := RowDimension(m): Matrix(1, dim, [seq(max(seq(m[i, j], j = 1 .. dim)), i = 1 .. dim)]) end proc: > ECI := proc (n) options operator, arrow: MatrixMatrixMultiply(RS(a(n)), Transpose(MX(d(n))))[1, 1] end proc: seq(ECI(n), n = 1 .. 77);
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, May 08 2012
STATUS
approved