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Abstract:The PI polynomial of a molecular graph is defined to be the sum X |E(G)|−N(e) + |V(G)|(|V(G)|+1)/2 − |E(G)| over all edges of G, where N(e) is the number of edges parallel to e. In this paper, the PI polynomial of the phenylenic nanotubes and nanotori are computed. Several open questions are also included.

In this paper, we give some equivalent conditions for Lie algebras to be isoclinic. In particular, it is shown that if two Lie algebras L and K are isoclinic then L can be constructed from K and vice versa using the operations of forming direct sums, taking subalgebras, and factoring Lie algebras. We also study connection between isoclinic and the Schur multiplier of Lie algebras. In addition, we deal with some properties of covers of Lie algebras whose Schur multipliers are finite dimensional and prove that all covers of any abelian Lie algebra have Hopfian property. Finally, we indicate that if a Lie algebra L belongs to some certain classes of Lie algebras then so does its cover.