The algebraic approach to the construction of the reflexive polyhedra thatyield Calabi-Yau spaces in three or more complex dimensions with K3 fibresreveals graphs that include and generalize the Dynkin diagrams associated withgauge symmetries. In this work we continue to study the structure of graphsobtained from $CY_3$ reflexive polyhedra. We show how some particularly definedintegral matrices can be assigned to these diagrams. This family of matricesand its associated graphs may be obtained by relaxing the restrictions on theindividual entries of the generalized Cartan matrices associated with theDynkin diagrams that characterize Cartan-Lie and affine Kac-Moody algebras.These graphs keep however the affine structure, as it was in Kac-Moody Dynkindiagrams. We presented a possible root structure for some simple cases. Weconjecture that these generalized graphs and associated link matrices maycharacterize generalizations of these algebras.
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