  We review the Batyrev approach to Calabi-Yau spaces based on reflexive weight
vectors. The Universal CY algebra gives a possibility to construct the
corresponding reflexive numbers in a recursive way. A physical interpretation
of the Batyrev expression for the Calabi-Yau manifolds is presented. Important
classes of these manifolds are related to the simple-laced and
quasi-simple-laced numbers. We discuss the classification and recurrence
relations for them in the framework of quantum field theory methods. A relation
between the reflexive numbers and the so-called Berger graphs is studied. In
this correspondence the role played by the generalized Coxeter labels is
highlighted. Sets of positive roots are investigated in order to connect them
to possible new algebraic structures stemming from the Berger matrices.

High Energy Physics - Theory;Algebraic Geometry;Group Theory