Graph Theory and Spectral Effects in CRAM and BDF Solutions of the Nuclear Inventory Equation

Nuclear data consist of nuclide-specific files that tabulate a wide range of quantities, including reaction cross sections, scattering and absorption probabilities, and decay coefficients. Here we reinterpret this body of data through a graph-theoretic lens, representing it algebraically as an adjacency matrix. We then analyze how the structural and spectral properties of this matrix influence numerical solutions of the nuclear inventory equation, with particular emphasis on the Chebyshev rational approximation method and the backward differentiation formula methods.Activation-decay simulations of nuclear inventories typically proceed in two stages: an activation phase under irradiation, followed by a decay phase after the source is removed. While both phases solve the same stiff system of linear ordinary differential equations, the decay phase has a distinctive property: the transmutation graph becomes acyclic. We exploit this by applying a topological ordering of isotopes, which transforms the decay matrix into strictly triangular form. This removes the need for expensive LU factorizations during the decay phase, replacing it with a single forward or back-substitution step. The result is a gain in computational speed with increased accuracy.