Reflexive Numbers and Berger Graphs from Calabi-Yau Spaces

A novel relation between number theory and recently found Berger graphs is studied. The Berger graphs under investigation are constructed mainly for the CY_3 space. The method of analysis is based on the slice classification of CY_l polyhedra in the so-called Universal Calabi-Yau algebra. The concept of reflexivity in these polyhedra is reviewed and translated into the theory of reflexive numbers. A new approach based on recurrence relations and Quantum Field Theory methods is applied to the simply-laced and quasi-simply-laced subsets of the reflexive numbers. In the correspondence between the reflexive vectors and Berger graphs the role played by the generalized Coxeter labels is shown to be important. We investigate the positive roots of some of the Berger graphs to guess the algebraic structure hidden behind them.