**4. Main Theorem**

We consider the case study on *G*0 by

**Case A:** *A*0,*σ* = ∅ and *G*0 is a bipartite graph;

**Case B:** *A*0,*σ* = ∅ and *G*0 is a non-bipartite graph;

**Case C:** *A*0,*σ* = ∅ and *G*0 \ *A*0,*σ* is a bipartite graph;

**Case D:** *A*0,*σ* = ∅ and *G*0 \ *A*0,*σ* is a non-bipartite graph.

For a subspace H ⊂ C*A*0 , the projection operator onto H is denoted by Π H. Then, we obtain the following theorem.

**Theorem 1.** *Let φn be the nth iteration of the Grover walk on G* = ( *V*, *A*) *with sinks. Let the survival probability at time n be defined by*

$$\gamma\_n = \sum\_{a \in A} |(\phi\_n)|^2.$$

*The subspaces* A, B, C, D *of* C*A*0 *are defined in (7),..., (10), respectively. Then, we have*

*1.* lim*n*<sup>→</sup> ∞ *γn* = *γ exists.*

*2. The survival probability γ is expressed by*

$$\gamma = ||\Pi\_{\mathcal{T}} \chi\_S \phi\_0||^2 + ||\Pi\_{\mathcal{K}} \chi\_S \phi\_0||^2 + \begin{cases} ||\Pi\_{\mathcal{A}} \chi\_S \phi\_0||^2 & \colon \text{Case } A \\ ||\Pi\_{\mathcal{B}} \chi\_S \phi\_0||^2 & \colon \text{Case } B \\ ||\Pi\_{\mathcal{C}} \chi\_S \phi\_0||^2 & \colon \text{Case } C \\ ||\Pi\_{\mathcal{D}} \chi\_S \phi\_0||^2 & \colon \text{Case } D \end{cases}$$

**Proof.** Part 1 of Theorem 1 is obtained by the consequences of Proposition 3 and Part 2 derives from Propositions 5 and 6.

From this theorem, we obtain useful sufficient conditions for non-zero survival probability as follows.

**Corollary 1.** *Assume G*0 *is a finite and connected graph. If G*0 *is not a tree or G*0 *has more than two self-loops, then γ* > 0*.*

**Remark 1.** *The eigenspaces* A, B, C, D *correspond to the p-attractors defined in [8].*
