Consider an electric circuit with dc sources ( voltage and current) and resistors. Write down the equations. In the most general case, the solution of the system is not unique. The set of solutions can be empty or positive dimensional (simple example: 2 points in the graph and two batteries in parallel joining the points).
The dimension of the space of solutions can be computed with two different methods: 1) Mathematically : compute the determinants and the compatibility conditions 2) Physically. Give conditions on the graph to give a meaning for the dimension. For instance a cycle of batteries give an empty set of solutions or make the dimension increase by one.
I am interested in second method and I am looking for references in the litterature.
What I found is the following. In the student books, the unicity is always assumed to be true, eg. in the standard book by nilsson and riedel. In more advanced books, I have found discusions only for particular cases, and with very technical tools. For instance, in Frankel (The geometry of physics), only purely resistive circuits are considered with source of currents in the nodes. And the proof uses ( a simplified version of) Hodge theory.
Now my question: - Is there a book or an article where the very general case ( any graph with dc sources and resistors ) is considered and the dimension of the system described in terms of the graph ? I am interested both in sophisticated answers as above and in answers with basic tools of linear algebra. All references welcome.
I am looking for references, not for the solution. I have already written a solution for my students (an elementary one, with basic linear algebra). I want to compare my solution with the existing litterature. If this is useful and not a waste of time, I will make public my personal notes.