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In this article, we introduce an algorithmic method to find the conservative energy and non-conservative power of a large class of maximally nonlinear electric circuits (including Josephson tunnel junctions, coherent quantum phase slips, and superconducting loops), based on the incidence matrix of the circuits' digraph. We consider two-port linear circuits with mostly holonomic constraints provided by either Maxwell-Kirchhoff's current rules or Maxwell-Kirchhoff's voltage rules. The circuit's independent variables, generally a superset of the degrees of freedom, are obtained from the solution space of Maxwell-Kirchhoff's current or voltage rules. The method does not require to find any Lagrangian. Instead, the circuit's classical or quantum Hamiltonian is obtained from the energy of the reactive (i.e., conservative) circuit elements by means of transformations complementary to Hamilton's equations. Dissipation (loss) is accounted for by using the Rayleigh dissipation function and defining generalized Poisson brackets--Poisson-Rayleigh brackets. Fluctuations (noise) are added as voltage or current sources characterized by bath modes. Non-conservative elements (e.g., noisy resistors) are included ab initio using the incidence-matrix method, without needing to treat them as separate elements. Finally, we show that in order to form a complete set of canonical coordinates, auxiliary (which could be parasitic in certain cases) circuit elements are required to find the Hamiltonian of circuits with an incomplete set of generalized velocities. In particular, we introduce two methods to eliminate the coordinates associated with the auxiliary elements by either Hamiltonian reduction or equation-of-motion reduction. We use auxiliary circuit elements to treat a maximally nonlinear circuit comprising simultaneously both a Josephson junction and a quantum phase slip.