“…There have been many advances in Wiener index, Szeged index, PI index, and other degree-based or distance-based indices of molecular graphs, while the study of the redefined Zagreb indices of nano structures has been largely limited. Furthermore, nanotube, nanostar, polyomino chain and benzenoid series are critical and widespread molecular structures which have been widely applied in medical science, chemical engineering and pharmaceutical fields (see Mirzargar [18], Ashrafi and Karbasioun [1], Manuel et al, [16], Baca et al, [4], DeBorde et al, [5], Matsuno et al, [17], Vilela et al, [22], Velichko, Nosich [21], Haslam and Raeymaekers [12], Ashrafi and Karbasioun [2]). Also, these structures are the basic and primal structures of other more complicated chemical molecular structures and nano materials.…”