What exactly are electrical circuits as mathematical objects?

It seems quite intuitive to me, that they are geometric realization of some graph with some additional structure.

Another thing I notice is that application of KVL and KCL together turns this graph to a system of linear equations.

$$\text{circuit} \to^{KVL}_{KCL} \to \text{linear eqtns}$$

All sites I look online eg: Wikipedia give me undetailed answers.

Edit: I see that many people are like there is no singular view.... sure that's understandable... but sometimes it is very clear that some matters are much more clearer and understandable in certain views than others. For example, consider how differential forms give clarity on the whole bunch of vector calculus theorems, and whose mathematical properties quite directly suggest effects like that of Aharonov-Bohm, which would have been otherwise difficult to arrive at mathematically with standard theories of vector calculus.

Also, for the camp of people there who say it's just differential equations and see the right side of the above equation. I think they've not seen circuits enough. There are very many cases where the circuit itself offers quick ways to arrive at simplified equations.

Consider for example things like the Source Transformation Theorem's or simplifying complicated circuits using symmetry arguements