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](https://en.wikipedia.org/wiki/Electronic_circuit) give me undetailed answers.