Shape analysis of random polymer networks.

We propose the model of a random polymer network, formed on the base on Erdös-Rényi random graph. In the language of mathematical graphs, the chemical bonds between monomers can be treated as vertices, and their chemical functionalities as degrees of these vertices. We consider graphs with fixed number of vertices= 5 and variable parameter(connectedness), defining the total number of links=(- 1)/2 between vertices. Each link in such graphs is treated as a Gaussian polymer chain. The universal rotationally invariant size and shape characteristics, such as averaged asphericity and size ratio of such structures are obtained both numerically by application of Wei's method and analytically within the continuous chain model. In particular, our results quantitatively indicate an increase of asymmetry of polymer network structure when its connectednessdecreases.