symbolic circuit analysis in Sagemath/Mathematica with graph theory/incidence matrix?
There are several techniques to write the equations describing the behavior of a circuit. Topology-based techniques make use of results from graph-theory. They may be systematized for passive circuits, but the inclusion of active elements is not straightforward. The tableau approach is a systematic technique does not have these limitations but yields a high number of equations (and unknowns). In the modified nodal analysis (MNA) approach, the main unknowns are the nodal voltages (the name comes from this fact) augmented with the currents flowing across certain elements: the voltage sources and inductors. In the MNA, the system of equations is such that there are no negatives powers of s (the Laplace transform variable) in the system matrix. Most, if not all, of the current circuit simulators (for instance, all SPICE variants) are based on MNA as it a) is straightforward to apply to any circuit and b) gives a reasonable number of unknowns.

Symbolic analysis is a different question, and can be based on any analysis technique. For instance, symbolic computation of the determinants arising when solving the MNA system of equations directly gives the the transfer function in symbolic form.