I read Kirchhoff's original article from 1847 and tried to generalize his result to the case of the transcient regime of $R$, $C$ networks. The original article is limited to a network of $n$ resistors, denoted $w_i$ (which form the branches of the network) and voltage sources.
Here is the English translation of Kirchhoff's conclusion:
Let $m$ be the number of crossing points, i.e., the number of points at which two or more wires meet, and let $\mu = n - m + 1$. Then the common denominator of all quantities $I$ is the sum of all $w_{k_1}w_{k_2}….w_{k_\mu}$ for each $\mu$ elements of $w_{k_1}$, $w_{k_2}$….$w_{k_n}$ having the property that no closed figure remains after removal of the wires $ k_1$, $k_2$….$k_{\mu}$
I would have liked to use this result in the case of a network made up of resistors and capacitors. Using the Laplace transform, we are led to assign to each branch of the network a Laplace impedance $Z_i(x)=w_i+G_i/p =w_i+G_i x $ if we set $x=1/p$. We will have $ w_i \ge 0$ and $ G_i \ge 0$ Following Kirchhoff result, we should have :
The common denominator of all quantities $I$ is the sum of all $Z_{k_1}Z_{k_2}….Z_{k_\mu}$ for each $\mu$ elements of $Z_{k_1}$, $Z_{k_2}$….$Z_{k_n}$ having the property that no closed figure remains after removal of the wires $ k_1$, $k_2$….$k_{\mu}$.
This denominator is a polynomial in $x$ : $P(x)$.
Using the inverse Laplace transform, it appears that the roots $x_i$ of this polynomial brought to the variable $p$, $p_i=1/x_i$, will intervene in the transient regime in the form of exponentials $e^{ p_it}$ and for a network formed of resistances and capacitances, it seems clear that the $p_i$ must be real and negative. The polynomial $ P(x)$ must therefore have only negative real roots.
I am well aware that Kirchhoff's formulation is obsolete and that it is of course possible to formulate these results using the concepts of graph theory of which Kirchhoff is often considered one of its inventors. I understood that the sum of terms indicated by Kirchhoff is the determinant of the system of linear equations and the sum is a sum over all spanning trees. But, as a physicist, I don't have the energy to study graph theory in detail.
So, to resume, we have an electrical network with $n$ edges and $m$ vertices. We set $\mu = n - m + 1$. At each edge $i$, we associate an impedance $Z_i(x) =w_i+G_i x $ (with $ w_i \ge 0$ and $ G_i \ge 0$) and we consider the polynomial which is the sum of all the products $Z_{k_1}(x)Z_{k_2}(x)….Z_{k_\mu}(x)$ for each $\mu$ elements of $Z_{k_1}$, $Z_{k_2}$….$Z_{k_n}$ having the property that no closed figure remains after removal of the wires $ k_1$, $k_2$….$k_{\mu}$.
Can anyone help me prove that the roots of this polynomial $P(x)$ are all real (it is obvious that if they are real, they are negative)