Shape analysis of random polymer networks

We analyze conformational properties of branched polymer structures, formed on the base of Erd\"os-R\'enyi random graph model. We consider networks with $N=5$ vertices and variable parameter $c$, that controls graph connectedness. The universal rotationally invariant size and shape characteristics, such as averaged asphericity $\langle A_3 \rangle$ and size ratio $g$ 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 connectedness $c$ decreases.