- Dual Operators

Next, we want to define a dual boundary operator for *∂*. Let *U* and *W* be finite-dimensional vector spaces, and consider a linear map *f* : *U* → *W*. Then, a *dual map f* ∗ : *W*∗ → *U*∗ is defined as [ *f* <sup>∗</sup>(*α*)](*u*) := *α*[ *f*(*u*)] for *α* ∈ *W*∗ and all *u* ∈ *U*. Let {*<sup>u</sup>*1, *u*2, ··· , *ul*} and {*<sup>w</sup>*1, *w*2, ··· , *wm*} be the bases of *U* and *W*, respectively. We have *f* <sup>∗</sup>(*w<sup>i</sup>*) = <sup>∑</sup>*lj*=<sup>1</sup>[ *f* <sup>∗</sup>(*w<sup>i</sup>*)](*uj*)*u<sup>j</sup>* = <sup>∑</sup>*lj*=<sup>1</sup> *w<sup>i</sup>*[ *f*(*uj*)]*u<sup>j</sup>* = <sup>∑</sup>*lj*=<sup>1</sup> *ujMij* with the matrix representation of *f* as *Mij* = *w<sup>i</sup>*[ *f*(*uj*)]. This shows that the matrix representation of the dual map is the transpose of the matrix representation of the original map. Using the concept of the dual map, we can define a *coboundary* operator d = *∂*∗ to satisfy *∂c ϕ* = *c* d*ϕ* for all *c* ∈ *C*1 and *ϕ* ∈ *C*<sup>0</sup> = (*<sup>C</sup>*0)<sup>∗</sup>. For a dual basis {*n<sup>i</sup>*} ⊂ *C*<sup>0</sup> obtained from 0-cells N = {*ni*}, we have d*n<sup>i</sup>* = ∑|E| *j*=1 *ej*Δ*ij* with the incidence matrix <sup>Δ</sup>*ij*. For the example of Figure 1b, we have

$$\operatorname{cl}[n^1 \ n^2 \ n^3] := [\operatorname{cl} n^1 \ \operatorname{d} n^2 \ \operatorname{d} n^3] = [\operatorname{e}^1 \ \operatorname{e}^2 \ \operatorname{e}^3 \ \operatorname{e}^4] \begin{bmatrix} -1 & 1 & 0 \\ 0 & -1 & 1 \\ 0 & -1 & 1 \\ 1 & 0 & -1 \end{bmatrix} . \tag{3}$$

#### - Potential and Kirchhoff's Voltage Law

Finally, we discuss a relation between KVL and a potential. KVL was formulated to state that the voltage drop along any loop is zero, but how is this statement related to the existence of a potential? To see this, we start from a general statement. Consider a linear map *f* : *U* → *W*. The image and kernel of linear maps *f* and *f* ∗ are related through (ker *f*)<sup>⊥</sup> = im *f* ∗ and (im *f*)<sup>⊥</sup> = ker *f* ∗ [61]. The proof of the first statement is as follows. First, (ker *f*)<sup>⊥</sup> ⊃ im *f* ∗ holds because we have *f* <sup>∗</sup>(*β*)(*u*) = *β*(*f*(*u*)) = 0 for all *u* ∈ ker *f* with *β* ∈ *W*<sup>∗</sup>. Second, dim(ker *f*)<sup>⊥</sup> = dim *U* − dim ker *f* = rank *f* = rank *f* ∗ holds due to the rank–nullity theorem. Then, we obtain (ker *f*)<sup>⊥</sup> = im *f* ∗. A similar proof is applied for the second statement. Now, we come back to circuit theory and define *B*<sup>1</sup> = im d ⊂ *C*1. From (ker *∂*)<sup>⊥</sup> = im d, we obtain (*<sup>Z</sup>*1)<sup>⊥</sup> = *B*1. This means that there is a potential *ϕ* ∈ *C*<sup>0</sup> which satisfies *V* = −d*ϕ* for a voltage distribution *V*.

#### - Summary of Circuit Equations

The discussions so far show the duality between KCL and KVL. These results are summarized in Figure 4. Importantly, the degree of freedom for currents and voltages constrained by KCL and KVL is given by dim *Z*1 + dim *B*<sup>1</sup> = dim *C*1. On the other hand, a circuit element along each edge gives the relation between the current and voltage on the edges, and we have dim *C*1 equations with respect to all the circuit elements. Therefore, the current and voltage distributions are unambiguously determined.

**Figure 4.** Duality between Kirchhoff's current and voltage laws.

#### *2.2. Planar Graph as Cellular Paving*

Consider the series and parallel resistors shown in Figure 5. Series resistors *R*1 and *R*2 have the composite resistance *R* = *R*1 + *R*2. On the other hand, parallel resistors *R*1 and *R*2 have the total resistance *R*, satisfying 1/*R* = 1/*R*1 + 1/*R*2. We can clearly see the duality between resistance and conductance (given by an inverse relationship) for series and parallel resistances. This duality universally holds in more general situations, as we show in Section 2.5. In this subsection, we set up a fundamental geometric structure to establish circuit duality.

**Figure 5.** (**a**) series and (**b**) parallel resistors with composite resistances *R* and *R*, respectively. The duality between resistance and conductance appears as *R* = *R*1 + *R*2 and 1/*R* = 1/*R*1+ /*R*2.

#### -2-Chains in Planar Graphs

The graph shown in Figure 1b is *planar*, i.e., its edges intersect only at their nodes. For a planar graph, we can define faces. In Figure 6, we show directed faces F = { *f*1, *f*2, *f*3}, where the internal direction of each face is represented by the directed circle. Elements of F are called 2-cells. Note that we include the unbounded face *f*3 outside the circuit. The area of face *f*3 can be finite if we consider the planar graph to be on a sphere. The vector space generated by F is denoted by *C*2. It is natural to define a boundary operator *∂* : *C*2 → *C*1, such that *ej*(*<sup>∂</sup> fi*) = 1 [*ej*(*<sup>∂</sup> fi*) = −1] if *ej* ∈ E is included in *f* in the same [opposite] direction; otherwise, *ej*(*<sup>∂</sup> fi*) = 0. For Figure 6, we have

$$
\partial[f\_1 \ f\_2 \ f\_3] := [\partial f\_1 \ \partial f\_2 \ \partial f\_3] = [e\_1 \ e\_2 \ e\_3 \ e\_4] \begin{bmatrix} 1 & 0 & 1 \\ 1 & -1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix} . \tag{4}
$$

Note that *∂* ◦ *∂* = 0 holds, i.e., the boundary of a cell boundary is empty.

**Figure 6.** Faces in a planar circuit.
