- Boundary Operator

Now, we extract the connection information regarding G. Consider an edge *e* directing from a node *n*1 to *n*2. As shown in Figure 2, the boundary of *e* is given by *∂e* = *n*2 − *n*1. Extending the definition linearly, we can introduce a linear *boundary operator ∂* : *C*1 → *C*0. The matrix representation of *∂* is denoted by <sup>Δ</sup>*ij*, called the *incidence matrix*. For the previous example in Figure 1b, we obtain the following matrix representation of *∂*:

$$
\partial \begin{bmatrix} \varepsilon\_1 \ v\_2 \ c \gamma \ c\_4 \end{bmatrix} := \begin{bmatrix} \partial \varepsilon\_1 \ \partial \varepsilon\_2 \ \partial \varepsilon\_3 \ \partial \varepsilon\_4 \end{bmatrix} = \begin{bmatrix} n\_1 \ n\_2 \ n\_3 \end{bmatrix} \Delta \tag{1}
$$

with

$$
\Delta = [\Delta^i\_j] = \begin{bmatrix} -1 & 0 & 0 & 1 \\ 1 & -1 & -1 & 0 \\ 0 & 1 & 1 & -1 \end{bmatrix} \,. \tag{2}
$$

**Figure 2.** Action of the boundary operator. Here, we have *∂e* = *n*2 − *n*1.

#### - Kirchhoff's Current Law

To understand the physical meaning of *∂*, we consider *∂I* for a current distribution *I* ∈ *C*1. The coefficient of *∂I* for *n* ∈ N represents the net inflow of the current: the current inflow to *n* minus the current outflow from *n*. Therefore, the boundary operator can be used to express Kirchhoff's current law (KCL), which states that net current inflows at each node are zero. KCL restricts current distribution to a linear subspace *Z*1 = ker *∂* = {*c* ∈ *<sup>C</sup>*1|*∂<sup>c</sup>* = <sup>0</sup>}. For Equation (2), we have rank Δ = 2, and dim *Z*1 = dim *C*1 − rank Δ = 2 from the rank–nullity theorem [61]. The basis of *Z*1 is given by {*<sup>m</sup>*1 = *e*1 + *e*2 + *e*4, *m*2 = *e*2 − *<sup>e</sup>*3}. Here, *m*1 and *m*2 are closed loops called *meshes*. Generally, we can construct a basis of *Z*1 with meshes [60].
