The eigen-polynomials
of the Rouse matrix for the G-generation dendritic
chains with arbitrary spacer length (n), and functionalities
of the central segment (f
0) and outer
segments (f) are
derived with the aid of graph theoretical method. The exact expression
for the radius of gyration with arbitrary f and f
0 has also been obtained by using the relationships
between the coefficients and the roots of eigen-polynomial. For the
special case of the standard dendrimers with no spacer, i.e., f
0 = f and n = 0, the approximate eigenvalues are obtained and agree with the
numerical results very well. Based on the eigenvalues obtained, the
dynamic properties of this class of dendrimers, the intrinsic storage
moduli G′(ω) and loss moduli G″(ω), are calculated and compared to those
of linear and starlike chains. The graph theoretical method we developed
is useful for dealing with the polymer chains with complex topological
structures, especially suitable when the chain graph possesses certain
symmetric elements.