  The algebraic approach to the construction of the reflexive polyhedra that
yield Calabi-Yau spaces in three or more complex dimensions with K3 fibres
reveals graphs that include and generalize the Dynkin diagrams associated with
gauge symmetries. In this work we continue to study the structure of graphs
obtained from $CY_3$ reflexive polyhedra. The objective is to describe the
``simply laced'' cases, those graphs obtained from three dimensional spaces
with K3 fibers which lead to symmetric matrices. We study both the affine and,
derived from them, non-affine cases. We present root and weight structurea for
them. We study in particular those graphs leading to generalizations of the
exceptional simply laced cases $E_{6,7,8}$ and $E_{6,7,8}^{(1)}$. We show how
these integral matrices can be assigned: they may be obtained by relaxing the
restrictions on the individual entries of the generalized Cartan matrices
associated with the Dynkin diagrams that characterize Cartan-Lie and affine
Kac-Moody algebras. These graphs keep, however, the affine structure present in
Kac-Moody Dynkin diagrams. We conjecture that these generalized simply laced
graphs and associated link matrices may characterize generalizations of
Cartan-Lie and affine Kac-Moody algebras.
