Recently, polymers of complex chemical
connectivity expressed with graphs have been synthesized in experiments [1-3]. It
is indeed marvelous that even such polymers expressed with K_{3,3} bipartite graph have been produced [2]. We call
polymers with nontrivial structures in chemical connectivity *topological polymers*. We also call
polymers with nontrivial topology of spatial graphs as embeddings in three
dimensions* topological polymers* [4].

The Rouse dynamics of a polymer plays an important role in the dynamical aspects of the polymer in dilute solution, and also in melts if the molecular weight is small [5]. In this talk, we formulate the Rouse dynamics of the topological polymer with a given graph. We derive several physical consequences of the model. In particular, we compare the experimental data of Size Exclusion Chromatography (SEC) of some topological polymers with their theoretical estimates of the mean-square radius of gyration obtained by the Gaussian method [6]. We argue physical backgrounds of SEC data and discuss how they are consistent.

After reviewing the method for constructing Gaussian random configurations of a topological polymer, i.e. Gaussian random graph embeddings [6], we derive the normal coordinates and modes for the topological polymer. Here we remark that while the Moore-Penrose generalized inverse matrix has been addressed for general Gaussian molecules about three decades ago [7], it seems that important consequences were found later independently [8]. Moreover, it has not been known until quite recently how to generate Gaussian random configurations of a given topological polymer [6]. In fact, we can calculate any physical quantity at least numerically by taking the ensemble averages over generated configurations. It is quite nontrivial to generate such random walks that satisfy the constraints of all independent loops in the graph.

The results of this talk are obtained in collaboration with Jason Cantarella, Clayton Shonkwiler, and Erica Uehara.

*1)
**Topological Polymer Chemistry:
Progress of cyclic polymers in synthesis, properties and*

* functions*, Y. Tezuka ed., World
Scientific, Singapore, 2013.

2)
T. Suzuki, T. Yamamoto and Y. Tezuka. *J. Am. Chem.
Soc.*, 2014, **136**, 10148–10155.

3)
Y. Tezuka. *Acc. Chem. Res.*, 2017, **50**, 2661–2672.

4)
E. Uehara and T. Deguchi, *J. Chem.
Phys.* 2016, **145**, 164905.

5)
M. Doi and S. F. Edwards, *The Theory
of Polymer Dynamics*, Oxford University Press, Oxford, 1986.

6) J. Cantarella, T. Deguchi, C. Shonkwiler and E. Uehara, in preparation.

7)
B. E. Eichinger, *Macromolecules*,
1980, **13**, 1-11.

8) E. Estrada and N. Hatano, *Chem.
Phys. Let. *2010, **486**, 166–170.