Edge-Wiener index is a new topological invariant which is used in the study of physicochemical properties of chemical compounds and is closely associated with the Wiener index-To investigate the related properties of a class of important polygon chains comprehensively

an explicit expression for the expected value of the Edge-Wiener indices of a class of random polygonal chains is given. For applications

the Polyphenyl chains which can achieve the maximum and minimum of the EdgeWiener indices is found. And some results that have been published in recent papers are revealed. Finally the asymptotic behaviour of EdgeWiener indices is also considered.