OMS-Vol. 2 (2018), Issue 1, pp. 351–360 Open Access Full-Text PDF

Abstract:Let \(k>0\) an integer. \(F\), \(\tau \), \(N\), \(N_{k}\), \(N_{k}^{(2)}\) and \(A\) denote the classes of finite, torsion, nilpotent, nilpotent of class at most \(k\), group which every two generator subgroup is \(N_{k}\) and abelian groups respectively. The main results of this paper is, firstly, we prove that, in the class of finitely generated \(\tau N\)-group (respectively \(FN\)-group) a \((FC)\tau \)-group (respectively \((FC)F\)-group) is a \(\tau A\)-group (respectively is \(FA\)-group). Secondly, we prove that a finitely generated \(\tau N\)-group (respectively \(FN\)-group) in the class \(((\tau N_{k})\tau ,\infty)\) (respectively \(((FN_{k})F,\infty)\)) is a \(\tau N_{k}^{(2)}\)-group (respectively \(FN_{k}^{(2)}\)-group). Thirdly we prove that a finitely generated \(\tau N\)-group ( respectively \(FN\)-group) in the class \(((\tau N_{k})\tau ,\infty)^{\ast}\) (respectively \(((FN_{k})F,\infty)^{\ast}\)) is a \(\tau N_{c}\)-group (respectively \(FN_{c}\)-group) for certain integer \(c\) and we extend this results to the class of \(NF\)-groups.

OMS-Vol. 2 (2018), Issue 1, pp. 338–350 Open Access Full-Text PDF

Abstract:The generalized hierarchical product of graphs was introduced by L. Barrière et al. in 2009. In this paper, reformulated first Zagreb index of generalized hierarchical product of two connected graphs and hence as a special case cluster product of graphs are obtained. Finally using the derived results the reformulated first Zagreb index of some chemically important graphs such as square comb lattice, hexagonal chain, molecular graph of truncated cube, dimer fullerene, zig-zag polyhex nanotube and dicentric dendrimers are computed.