Star-Shaped Integral Cartan-Type Matrices and an Egyptian-Fraction Classification of Affine Weighted Trees

Abstract

We study a concrete family of symmetric integral $Z$-matrices attached to weighted star trees. The arms are ordinary type-$A$ chains and the central diagonal entry is an arbitrary positive integer $k$ rather than being fixed to the Cartan value $2$. This gives a matrix-theoretic and graph-theoretic version of the so called Berger construction: it extends the simply laced affine Dynkin stars while remaining accessible through elementary linear algebra. For a star with arm lengths $r_1,\dots ,r_m$ we compute the determinant, the inertia, the positive-definite and affine regimes, and the primitive positive null vector in the affine case. The affine condition is exactly the unit-fraction equation \[ \sum_{i=1}^m \frac{1}{r_i+1}=m-k, \] so the classification of these affine weighted trees reduces to a finite Egyptian-fraction enumeration for each fixed pair $(m,k)$. The classical affine diagrams $D_4^{(1)}$, $E_6^{(1)}$, $E_7^{(1)}$ and $E_8^{(1)}$ appear as small subfamilies, while higher-arm cases give new integral positive-semidefinite star matrices with explicit Coxeter labels.