**Proposition 3.**


**Proof.** The inflow *ρ* = *χ*<sup>∗</sup>*Uψ*0 is orthogonal to H*c* by a direct consequence of Lemma 3.5 in [19], which implies *Enρ* ∈ H*s* for any *n* ∈ N by Proposition 2. Since the stationary state of Part 1 is described by the limit of the following recurrence

$$
\chi\_T \psi\_n = E \chi\_T \psi\_{n-1} + \rho\_\prime \quad \chi\_T \psi\_0 = 0,
$$

we obtain the conclusion of Part 1. On the other hand, let us consider the proof of Part 2 in the following. The time evolution in *G*0 obeys *χSφn* = *EχSφ<sup>n</sup>*−1. The overlap of *χSφn* to the space H*s* decreases more quickly than polynomial times because all the absolute values of the generalized eigenvalues of H*s* are strictly less than 1 (see Proposition 4 for more detailed order of the convergence). Then, only the contribution of the centered eigenspace, whose eigenvalues lie on the unit circle in the complex plain, remains in the long time limit.

Let *W* = *PcE* = *EPc* = *PcEPc* be the operator restricted to the centered eigenspace H*<sup>c</sup>*. Then, we have

$$\lim\_{n \to \infty} |\chi\_S \phi\_n(a) - \mathcal{W}^n \chi\_S \phi\_0(a)| = 0$$

for any *a* ∈ *A*0 uniformly by Proposition 3. This means that, in the long time limit, the time evolution is reduced to *W*, which is a unitary operator on H*<sup>c</sup>*.

**Proposition 4.** *The survival probability is re-expressed by*

$$\gamma = \| |P\_{\mathbf{c}} \chi\_S \phi\_0| \|^2 \text{.}$$

*The convergence speed (f*(*n*) = *<sup>O</sup>*(*g*(*n*)) *means* lim*n*→∞ | *f*(*n*)/*g*(*n*)| < ∞ *if the limit exists) is estimated by <sup>O</sup>*(*nκrnmax*)*, where κ* = dim H*s, rmax* = max{|*λ*| ; *λ* ∈ Spec(*E*), |*λ*| < <sup>1</sup>}*.*

**Proof.** Putting *E*(1 − *Pc*) = *W*, we have

$$\mathcal{W} + \mathcal{W}' = E, \quad \mathcal{W}\mathcal{W}' = 0,$$

by Proposition 2 (2). Note that the operator *En* is similar to

$$\bigoplus\_{\lambda \in \text{Spec}(E)} J^n(\lambda; k\_\lambda)$$

with some natural numbers *kλ*s. Here, *J*(*<sup>λ</sup>*; *k*) is the *k*-dimensional matrix by

$$J(\lambda;k) = \begin{bmatrix} \lambda & 1 & & & \\ & \lambda & 1 & & \\ & & \ddots & \ddots & \\ & & & \ddots & 1 \\ & & & & \lambda \end{bmatrix}.$$

We obtain that the survival probability at each time *n* is described by

$$\begin{aligned} \gamma\_{\mathbb{H}} &= ||\mathcal{U}\_G \chi\_S^\* E^{\mathbb{H}-1} \chi\_S \phi\_0 ||^2 \\ &= ||\mathcal{U}\_G \chi\_S^\* (\mathcal{W}^{n-1} + \mathcal{W}^{\prime n-1}) \chi\_S \phi\_0 ||^2 \\ &= ||(\mathcal{W}^{n-1} + \mathcal{W}^{\prime n-1}) \chi\_S \phi\_0 ||^2 \\ &= ||\mathcal{W}^{n-1} \chi\_S \phi\_0 ||^2 + ||\mathcal{W}^{\prime \prime n-1} \chi\_S \phi\_0 ||^2. \end{aligned}$$

In the third equality, we use the fact that *UG* is unitary; the last equality follows from Corollary 3. The second term decreases to zero by Proposition 2 (2) with the convergence speed at least *<sup>O</sup>*(*nκrnmax*) because the Jordan matrix *J*(*<sup>λ</sup>*; *k*) can be estimated by *J*(*<sup>λ</sup>*; *k*)*n* = *<sup>O</sup>*(*n<sup>k</sup>*|*λ*|*n*). Hence, we find for *γn*

$$\begin{aligned} \gamma\_n &= ||\mathcal{W}^{n-1} \chi\_S \phi\_0 ||^2 + O(n^\kappa r\_{\text{max}}^\mathfrak{n}) \ (n &> 1), \\ &= ||\mathcal{W}^{n-1} P\_\mathfrak{c} \chi\_S \phi\_0 ||^2 + O(n^\kappa r\_{\text{max}}^\mathfrak{n}) \\ &= ||P\_\mathfrak{c} \chi\_S \phi\_0 ||^2 + O(n^\kappa r\_{\text{max}}^\mathfrak{n}), \end{aligned}$$

where in the second equality we use that *W* = *WPc* and the last equality follows from Proposition 2 (3).

Therefore, the characterization of H*c* is important to obtain the asymptotic behavior of *φ<sup>n</sup>*.

### *7.2. Characterization of Centered Generalized Eigenspace by Graph Notations*

The centered generalized eigenspace of *E* can be rewritten by using the boundary operator *d*1 and the self-adjoint operator *T* = *<sup>d</sup>*1*Sd*<sup>∗</sup>1as follows.

**Lemma 3** ([19])**.** *Assume λ* ∈ Spec(*E*) *with* |*λ*| = 1*. Then, we have*


In the following, we consider the characterization of ker(±1 − *E*) using some walks on graph *G*0 up to the situations of the graph (Cases (A)–(D)). First, we prepare the following notations. For each support edge *e* ∈ *E*0, there are two arcs *a* and *a* such that |*a*| = |*a*|. Let us choose one of the arcs from each *e* ∈ *E*0 and denote *A*+ as the set of selected arcs. Then, |*<sup>A</sup>*+| = |*<sup>E</sup>*0| and *a* ∈ *A*+ if and only if *a* ∈/ *A*+ holds. We set *Arep* = *A*0,*σ* ∪ *A*+. Let us introduce the map *ι* : C*A*0 → C*Arep* defined by (*ιψ*)(*a*) = *ψ*(*a*) for any *ψ* ∈ C*A*0 and *a* ∈ *Arep*.

Let us define the boundary operator *∂*+ : C*Arep* → C*V*0 by

$$(\partial\_+\varphi)(u) = \sum\_{\substack{t(a) = u \ \text{in } A\_+}} \varphi(a) - \sum\_{\substack{o(a) = u \ \text{in } A\_+}} \varphi(a)$$

for any *ϕ* ∈ C*Arep* and *u* ∈ *V*0. On the other hand, let us also define the boundary operator *∂*− : C*Arep* → C*V*0 by

$$(\partial\_-\varphi)(u) = \begin{cases} \sum\_{\substack{t(a) = u \\ t(a) = u}} \varphi(a) + \sum\_{\substack{\varrho(a) = u \\ \varrho(a) = u}} \varrho(a) & : u \text{ has no selfloop,} \\ \sum\_{\substack{\varrho(a) = u \\ \varrho(a) = u}} \varphi(a) + \sum\_{\substack{\varrho(a) = u \\ \varrho(a) = u}} \varphi(a) - \varphi(a\_s) & : u \text{ has a selfloop } a\_{s'} \end{cases}$$

for any *ϕ* ∈ C*Arep* and *u* ∈ *V*0. We obtain the following lemma.

**Lemma 4.** *Let G*0 = (*<sup>V</sup>*0, *<sup>A</sup>*0) *be a graph with self-loops. We set E*0 *as the set of support edges of A*0 \ *A*0,*σ such that E*0 = {|*a*| | *a* ∈ *A*0 \ *<sup>A</sup>*0,*σ*}*. Then, we have*

$$\dim[\ker(1 - E)] = |E\_0| - |V\_0| + 1\_\prime$$

$$\dim[\ker(1+E)] = \begin{cases} |E\_0| - |V\_0| + 1 & \colon \text{Case } A\_\prime \\ |E\_0| - |V\_0| & \colon \text{Case } B\_\prime \\ |E\_0| - |V\_0| + |A\_{0\sigma}| & \colon \text{Case } \mathbb{C} \text{ and } D\_\prime \end{cases}$$

**Proof.** Note that, if *ψ* ∈ ker(1 + *<sup>S</sup>*), then *ψ*(*a*) = −*ψ*(*a*) for any *a* ∈ *A*+, and, if *ψ* ∈ ker(*d*), then ∑*t*(*a*)=*u ψ*(*a*) = 0 for any *u* ∈ *V*0. We remark that, since (*<sup>S</sup>ψ*)(*as*) = *ψ*(*as*) for any *as* ∈ *A*0,*<sup>σ</sup>*, we have *ψ*(*as*) = 0 if *ψ* ∈ ker(1 + *<sup>S</sup>*). Therefore, if *ψ* ∈ ker(1 + *S*) ∩ ker(*d*), then

$$\sum\_{\substack{\iota(a)=u \ \text{in}\ A\_{+}}} (\iota \psi)(a) - \sum\_{\substack{\iota(a)=u \ \text{in}\ A\_{+}}} (\iota \psi)(a) = (\partial\_{+} \iota \psi)(u) = 0$$

holds. Then, ker(1 + *S*) ∩ ker *d* is isomorphic to {*ϕ* ∈ ker *∂*+ | *supp*(*ϕ*) ⊂ *<sup>A</sup>*+}. Let us consider ker *∂*+. By the definition of *∂*+, we have *<sup>∂</sup>*+*δ*(*Arep*) *a* = 0 for any *a* ∈ *As*. Hence, we should eliminate the subspace of ker *∂*+ induced by the self-loops. The dimension of this subspace is |*<sup>A</sup>*0,*σ*|. The adjoint operator *∂*<sup>∗</sup>+ : C*V*0 → C*A*+ of *∂*+ is described by

$$(\partial\_+^\*f)(a) = f(t(a)) - f(o(a)),$$

for any *f* ∈ C*V*0 and *a* ∈ *Arep*. If *∂*<sup>∗</sup>+ *f* = 0 holds, then *f*(*t*(*a*)) = *f*(*o*(*a*)) for any *a* ∈ *A*+. This means *f*(*u*) = *c* for any *u* ∈ *V*0 with some non-zero constant *c*. Thus, dim ker(*∂*∗+) = 1. Therefore, the fundamental theorem of linear algebra (for a linear map *g* : *X* → *Y*, dim ker *g* = dim *X* − dim *Y* + dim ker *g*<sup>∗</sup>) implies

$$\begin{aligned} \dim \ker(\mathbf{1} + S) \cap \ker d &= \dim \ker(\partial\_{+}) - |A\_{0, \sigma}| \\ &= (|A\_{\text{rep}}| - |V\_{0}| + 1) - |A\_{0, \sigma}| \\ &= |E\_{0}| - |V\_{0}| + 1. \end{aligned}$$

Next, let us consider dim(ker(1 − *S*) ∩ ker *d*1). Note that, if *ψ* ∈ ker(1 − *<sup>S</sup>*), then *ψ*(*a*) = *ψ*(*<sup>a</sup>*). Assume that *ψ* ∈ ker(1 − *S*) ∩ ker(*d*1); then,

$$\sum\_{t(a)=u} \left(\iota\psi\right)(a) = 0 \text{ for any } u \in V\_{0\prime}$$

which is equivalent to

$$
\partial\_- \iota \psi = 0.
$$

The adjoint of *∂*− is described by

$$(\partial\_-^\*f)(a) = \begin{cases} f(t(a)) + f(o(a)) & : a \in A\_{+\prime} \\ f(t(a)) & : a \in A\_{0\rho} \end{cases}$$

Let us consider *f* ∈ ker(*∂*∗−) in the cases for both *A*0,*σ* = ∅ and *A*0,*σ* = ∅. *A*0,*σ* = ∅ **case**:

If *G*0 is a bipartite graph, then we can decompose the vertex set *V* into *X* ∪ *Y*, where every edge connects a vertex in *X* to one in *Y*. Then, *f*(*x*) = *k* for any *x* ∈ *X* and *f*(*y*) = −*k* for any *y* ∈ *Y* with some nonzero constant *k*. Hence, dim ker(*∂*∗) = 1 if *A*0,*σ* = ∅ and *G*0 is bipartite. On the other hand, if *G*0 is non-bipartite, then there must exist an odd length fundamental cycle *c* = (*<sup>a</sup>*0, *a*1,..., *<sup>a</sup>*2*m*). We have that

$$f(o(a\_1)) = -f(o(a\_2)) = f(o(a\_3)) = \dots = -f(o(a\_{2r})) = f(o(a\_0)) = -f(o(a\_1)).$$

Then, *f*(*u*) = 0 for any *u* ∈ *<sup>V</sup>*(*c*). Since *G*0 is connected, the value 0 is inherited to the other vertices by *f*(*t*(*a*)) = <sup>−</sup>*f*(*o*(*a*)). After all, we have *f* = 0, which implies ker(*∂*∗−) = 0 if *A*0,*σ* = ∅ and *G*0 is non-bipartite.

*A*0,*σ* = ∅ **case**: Since (*∂*∗− *f*)(*a*) = *f*(*t*(*a*)) = 0 if *a* ∈ *A*0,*<sup>σ</sup>*, then *f* takes the value 0 at the other vertices since *f*(*t*(*a*)) = −*f*(*o*(*a*)) for any *a* ∈ *A*+, which implies ker(*∂*∗+) = 0 if *A*0,*σ* = ∅.

After all, by the fundamental theorem of the linear algebra,

$$\dim \ker \partial\_- = |A\_{np}| - |V\_0| + \begin{cases} 1 & : A\_s = \oslash\_\bullet G\_0 \text{ is bipartite.}\\ 0 & : \text{otherwise.} \end{cases}$$

Noting that |*Arep*| = |*<sup>E</sup>*0| + |*<sup>A</sup>*0,*σ*|, we obtain the desired conclusion.

In the following, let us find linearly independent eigenfunctions of ker(±1 − *E*) using some concepts from graph theory. A walk *p* in *G*0 is a sequence *p* = (*<sup>a</sup>*0, *a*1, ... , *ar*) of arcs with *<sup>t</sup>*(*aj*) = *<sup>o</sup>*(*aj*+<sup>1</sup>) (*j* = 0, 1, ... ,*r* − 1), which may contain repeated arcs as defined in Section 2.1. We set {*<sup>a</sup>*0, *a*1, ... , *ar*} =: *<sup>A</sup>*(*p*), and similarly *<sup>A</sup>*(*p*) = {*<sup>a</sup>*0, ... , *ar*} as *multi* sets. ˜

We describe *ξ*(±)*p* : {*<sup>a</sup>*0,..., *ar*}∪{*<sup>a</sup>*0,..., *ar*} → {±1} by

$$
\tilde{\mathcal{G}}\_p^{(+)}(a) = \begin{cases} 1 & : a \in A(p), \\ -1 & : \overline{a} \in A(p), \end{cases}
$$

$$
\mathcal{G}\_p^{(-)}(a) = \begin{cases} 1 & : |a| \in \{ |a\_j| \mid j \text{ is even} \}, \\ -1 & : |a| \in \{ |a\_j| \mid j \text{ is odd} \}. \end{cases}
$$

Then, we set the functions *ξ*(±) *p* ∈ C*<sup>A</sup>* by

$$\mathcal{I}\_{p}^{(\pm)}(a) = \begin{cases} \sum\_{\substack{b \colon a=b \\ b}} \tilde{\xi}\_{p}^{(\pm)}(b) & : a \in A(p) \cup \overline{A}(p), \\ 0 & : \text{otherwise.} \end{cases} \tag{6}$$

Now, we are ready to show the following proposition for ker(1 − *<sup>E</sup>*). **Proposition 5.** *Let ξ*(+) *c be defined as (6). Then, we have*

$$\ker(1 - E) = \text{span}\{\xi\_c^{(+)} \mid c \in \Gamma\}.$$

**Proof.** By the definition of *ξ*(+) *c* , we have *ξ*(+) *c* ∈ ker *d*1 ∩ ker(1 − *<sup>S</sup>*), which implies *ξ*(+) *c* ∈ ker(1 − *E*) by Lemma 3. We show the linear independence of {*ξ*(+) *c* }*c*∈Γ. Let us set Γ = {*<sup>c</sup>*1, ... , *cr*} and *ξj* := *ξ*(+) *cj* (*j* = 1, ... ,*<sup>r</sup>*) induced by the spanning tree T ⊂ *G*. Assume that

$$
\beta\_1 \check{\varsigma}\_1 + \dots + \beta\_r \check{\varsigma}\_r = 0.
$$

Put *ar* ∈ *<sup>A</sup>*0(*cr*) ∩ (*<sup>A</sup>*0 \ *<sup>A</sup>*(T)). From the definition of the fundamental cycle, we have

$$
\beta\_1 \pounds\_1(a\_r) + \dots + \beta\_r \pounds\_r(a\_r) = \beta\_r = 0.
$$

In the same way, let *ar*−1 ∈ *<sup>A</sup>*(*cr*−<sup>1</sup>) ∩ (*<sup>A</sup>*0 \ *<sup>A</sup>*(T)); then,

$$
\beta\_1 \mathfrak{z}\_1(a\_r) + \dots + \beta\_{r-1} \mathfrak{z}\_{r-1}(a\_{r-1}) = \beta\_{r-1} = 0.
$$

Then, using it recursively, we obtain *β*1 = ··· = *βr* = 0, which means *ξj*<sup>s</sup> are linearly independent.

Then, dim(K) = |Γ| = |*<sup>E</sup>*0|−|*<sup>V</sup>*0| + 1. By Lemma 4, we reach the conclusion.

Define Γ*<sup>o</sup>*, Γ*e* ⊂ Γ as the set of odd and even length fundamental cycles. In the following, to obtain a characterization of ker(1 + *E*) = ker(1 − *S*) ∩ ker(*d*1), we construct the function *η<sup>x</sup>*,*<sup>y</sup>* ∈ ker(1 − *S*) ∩ ker(*d*1), which is determined by *x*, *y* ∈ *A*0,*σ* ∪ Γ*<sup>o</sup>*. The main idea to construct such a function is as follows. By the definition of *ξ*(−) *q* for any walk *q*, *ξ*(−) *q* ∈ ker(1 − *<sup>S</sup>*). This is equivalent to assigning the symbols "+" and "−" alternatively to each edge along the walk *q*. If the walk *c* is an even length cycle, then a symbol on each edge of *c* is different from the ones on the neighbor's edges; this means

$$\sum\_{t(a) = u} \xi\_t^{(-)}(a) = 0,$$

for every *u*. Then, *ξ*(−) *c* ∈ ker(*d*1) ∩ ker(1 − *S*) holds. On the other hand, if the walk *c* = (*b*1,..., *br*) is an odd length cycle, then a "frustration" appears at *u* := *<sup>o</sup>*(*b*1); i.e.,

$$\sum\_{\mathfrak{k}(a)=\mathfrak{u}} \xi\_c^{(-)}(a) = 2.$$

There are two ways to vanish this frustration: the first is to make a cancellation by another frustration induced by another odd cycle *c* and the second is to push the frustration to a self-loop. That is the reason the domains of *x* and *y* are *A*0,*σ* ∪ Γ*<sup>o</sup>*. We give more precise explanations of the constructions as follows. See also Figure 3.

**Figure 3. Construction of eigenfunction** *η<sup>x</sup>*,*<sup>y</sup>* ∈ C*A*0 : Each graph with signs ± represents the function *η<sup>x</sup>*,*y*. The support of *η<sup>x</sup>*,*<sup>y</sup>* is included in the arcs of each graphs. The signs are the return values of this function at each arcs. The return values of the inverse arcs are the same as the original arcs. The signs are assigned alternatively along the red colored walks. At each time where the walk runs through an arc, we take the sum of the signs; e.g., in the case for *x* ∈ *<sup>A</sup>*0,*<sup>σ</sup>*, *y* ∈ Γ*<sup>o</sup>*, the walk runs through the self-loop twice, and then the return value at the self-loops of the function is 1 + 1 = 2.

**Definition 2. Construction of** *η<sup>x</sup>*,*<sup>y</sup>* ∈ C*A*0 **:**

*The function η<sup>x</sup>*,*<sup>y</sup> is described by ξ*(−) *q induced by a walk depending on the indexes of x*, *y. In this paper, we consider four cases of the domains of x and y: (1) x* ∈ Γ*o, y* ∈ Γ*o; (2) x* ∈ *A<sup>σ</sup>, y* ∈ *Aσ; (3) x* ∈ *A<sup>σ</sup>, y* ∈ Γ*o; and (4) x* ∈ Γ*o, y* ∈ *A<sup>σ</sup>.*

*1. x* ∈ Γ*o, y* ∈ Γ*o case:*

> *If G*0 *is a bipartite graph, let us fix an odd length fundamental cycle c*∗ = (*<sup>a</sup>*0, ... , *ar*−<sup>1</sup>) ∈ Γ*o and pick up another c* ∈ Γ*o* = (*b*0, ... , *bs*−<sup>1</sup>)*. We set the following walk q and define the function on* C*A*0 *; ξ*(−) *q* =: *aac*∗<sup>−</sup>*c, induced by c*<sup>∗</sup>, *c* ∈ Γ*o:*


*Note that, by the definition of the fundamental cycle, the intersection c*0 ∩ *c is a path in Case (1). Since G*0 *is connected, there is a path connecting c*∗ *to c and we fix such a path for every pair of* (*<sup>c</sup>*<sup>∗</sup>, *c*) *in Case (2).*

*2. x* ∈ *Aσ and y* ∈ *Aσ case:*

> *If the number of self-loops* |*<sup>A</sup>σ*| ≥ 2*, let us fix a self-loop a*∗ *from Aσ and a path between a*∗ *to each a* ∈ *Aσ* \ {*a*∗}*. Let us denote the path between a*∗ *and a by p* = (*p*1, ... , *pt*)*. Then, we set the walk from a*∗ *to a by q* = (*<sup>a</sup>*<sup>∗</sup>, *p*1,..., *pt*, *a*) *and ξ*(−) *q* =: *η<sup>a</sup>*∗<sup>−</sup>*a.*

*3. x* ∈ *Aσ and y* ∈ Γ*o case:*

> *If* |*<sup>A</sup>σ*| ≥ 1 *and G* \ *Aσ is a non-bipartite graph, let us fix a self-loop a*∗ *and pick up an odd cycle c* = (*b*1, ... , *bt*) ∈ Γ*o; if the self-loop <sup>o</sup>*(*<sup>a</sup>*∗) ∈ *<sup>V</sup>*(*c*)*, we set the walk starting from a*∗ *visiting all the vertices <sup>V</sup>*(*c*) *and returning back to a*∗ *by q* = (*<sup>a</sup>*<sup>∗</sup>, *b*1, ... , *bt*, *<sup>a</sup>*∗)*; and, for <sup>o</sup>*(*<sup>a</sup>*∗) ∈/ *<sup>V</sup>*(*c*)*, let us fix a path p* = (*p*1, ... , *pt*) *between <sup>o</sup>*(*<sup>a</sup>*∗) *and <sup>o</sup>*(*b*1) *and set the walk starting from a*∗ *visiting all the vertices <sup>V</sup>*(*p*) ∪ *<sup>V</sup>*(*c*) *and returning back to a*∗*; q* = (*<sup>a</sup>*<sup>∗</sup>, *p*1,..., *pt*, *b*0 ..., *bt*, *p*¯*t*,..., *p*¯1, *<sup>a</sup>*∗)*. Then, we set ξ*(−) *q* =: *η<sup>a</sup>*∗,*c.*

*4. x* ∈ Γ*o and y* ∈ *Aσ case: Let us fix an odd length fundamental cycle c*∗ ∈ Γ*o* = (*b*1, ... , *bs*−<sup>1</sup>) *and pick up a self-loop a* ∈ *A<sup>σ</sup>. Let us set a short length path p between o*(*a*) *and <sup>o</sup>*(*b*1)*. Then, we consider the same walk q as in Case (3) and set ξ*(−) *q*=: *η<sup>c</sup>*∗,*a*.

By the construction, we have *η<sup>x</sup>*,*<sup>y</sup>* ∈ ker(1 − *S*) ∩ ker(*d*1). Using the function *η<sup>x</sup>*,*y*, we obtain the following characterization of ker(−<sup>1</sup> − *<sup>E</sup>*).

**Proposition 6.** *Let ξ*(−) *c be defined by (6) and η<sup>x</sup>*,*<sup>y</sup> be the above. Let us fix a*∗ ∈ *Aσ and c*∗ ∈ Γ*o. Then, we have*

$$\left\{ \text{span}\{ \xi\_{\mathcal{C}\_{\square}}^{(-)} \mid \mathcal{C} \in \Gamma \} \right.\\ \tag{1} \qquad \qquad \qquad \qquad \qquad \text{: Case (A), } \Gamma \text{ is} \\ \text{rank}\left( \mathcal{C} \right) \neq \text{rank}\left( \mathcal{C} \right)$$

$$\ker(1+E) = \begin{cases} \text{span}\{\mathcal{I}\_{\mathcal{C}}^{(-)} \mid c \in \Gamma\} & : \text{Case (A),} \\ \text{span}\{\mathcal{I}\_{\mathcal{C}}^{(-)} \mid c \in \Gamma\_{\mathcal{C}}\} \oplus \text{span}\{\eta\_{a\_{\*}-\mathcal{C}} \mid c \in \Gamma\_{\mathcal{O}} \mid \{c\_{\*}\}\} & : \text{Case (B),} \\ \text{span}\{\mathcal{I}\_{\mathcal{C}}^{(-)} \mid c \in \Gamma\} \oplus \text{span}\{\eta\_{a\_{\*}-a} \mid a \in A\_{0,\mathcal{F}} \mid \{a\_{\*}\}\} & : \text{Case (C),} \\ \text{span}\{\mathcal{I}\_{\mathcal{C}}^{(-)} \mid c \in \Gamma\_{\mathcal{C}}\} \oplus \text{span}\{\eta\_{a\_{\*}-y} \mid y \in \Gamma\_{\mathcal{O}} \cup \{A\_{0,\mathcal{F}}\} \mid a\_{\*}\}) & : \text{Case (D),} \end{cases}$$

**Proof.** We put

$$\mathcal{A} := \text{span}\{\mathbb{S}\_c^{(-)} \mid c \in \Gamma\},\tag{7}$$

$$\mathcal{B} := \text{span}\{\mathfrak{f}\_{\mathfrak{c}}^{(-)} \mid \mathfrak{c} \in \Gamma\_{\mathfrak{c}}\} \oplus \text{span}\{\mathfrak{y}\_{\mathfrak{c}\_\*-\mathfrak{c}} \mid \mathfrak{c} \in \Gamma\_{\mathfrak{o}} \mid \{\mathfrak{c}\_\*\}\},\tag{8}$$

$$\mathcal{C} := \text{span}\{\mathfrak{f}\_{\mathbf{c}}^{(-)} \mid \mathbf{c} \in \Gamma\} \oplus \text{span}\{\eta\_{a\_{\*}-a} \mid a \in A\_{0,\sigma} \mid \{a\_{\*}\}\},\tag{9}$$

$$\mathcal{D} := \text{span}\{\mathfrak{f}\_{\mathfrak{c}}^{(-)} \mid \mathfrak{c} \in \Gamma\_{\mathfrak{c}}\} \oplus \text{span}\{\eta\_{a\_{\*} - \mathfrak{y}} \mid \mathfrak{y} \in \Gamma\_{\mathfrak{o}} \cup \left(A\_{0, \mathcal{F}} \nmid \{a\_{\*}\}\right)\}\tag{10}$$

(see also Figure 4). From the construction of *η<sup>x</sup>*,*<sup>y</sup>* and *ξ*(−) *c* , the linear independence is immediately obtained. Let us check the dimensions for each case.

In Case (A),

$$\dim(\mathcal{A}) = |\Gamma| = |E\_0| - |V\_0| + 1.$$

In Case (B),

$$\dim(\mathcal{B}) = |\Gamma\_c| + (|\Gamma\_o| - 1) = |E\_0| - |V\_0|.$$

In Case (C),

$$\dim(\mathcal{C}) = |\Gamma| + (|A\_{0,\sigma}| - 1) = |E\_0| - |V\_0| + |A\_{0,\sigma}|.$$

In Case (D),

$$\dim(\mathcal{D}) = |\Gamma\_{\varepsilon}| + (|\Gamma\_{o}| - 1) + (|A\_{0,\sigma}| - 1) = |E\_{0}| - |V\_{0}| + |A\_{0,\sigma}|.$$

By Lemma 4, we reach the conclusion.

**Remark 5.** *"M* ⊕ *N" in Proposition 6 means that M and N are just complementary spaces; the orthogonality is not ensured in general.*

**Remark 6.** *If* |<sup>Γ</sup>*o*| = 1 *in Case (B), we have* B = span{*ξ*(−) *c* | *c* ∈ <sup>Γ</sup>*e*}*. If* |*<sup>A</sup>*0,*σ*| = 1 *in Case (C), we have* C = span{*ξ*(−) *c* | *c* ∈ <sup>Γ</sup>}*.*

**Remark 7.** *The subspace* D *can be re-expressed by*

$$\mathcal{D} = \text{span}\{\xi\_c^{(-)} \mid c \in \Gamma\_\varepsilon\} \oplus \text{span}\{\eta\_{c\_\* - y} \mid y \in (\Gamma\_o \backslash \{c\_\*\}) \cup A\_{0,\tau}\}.$$

**Figure 4. Eigenspaces** (A–D): This figure shows examples of four graphs for Cases (*A*)–(*D*) and their induced eigenspaces of the Grover walk (A–D). The figures at the right corner are the fundamental cycles for each case. The weighted graphs represent bases of each eigenspace. The weights are the return values at each arcs of the bases, where every base takes the value 0 at the dashed arcs.
