**8. Conclusions**

We investigated the Grover walk on a finite graph *G* with sinks using its connection with the walk on the graph *G*0 with tails. It was shown that the centered generalized eigenspace of the Grover walk with tails corresponds to the attractor space of the Grover walk with sinks, i.e., it contains all trapped states which do not contribute to the transport of the quantum walker into the sink. Consequently, the attractor space of the Grover walk with sinks can be characterized using the persistent eigenspace of the underlying random walk whose supports have no overlaps to the boundary and the concept of "flow" from graph theory. In particular, we constructed linearly independent basis vectors of the attractor space using the properties of fundamental cycles of *G*0. The attractor space can be divided into subspaces T and K, corresponding to the eigenvalues *λ* = ±1 and *λ* = 1, respectively, and an additional subspace which belongs to the eigenvalue *λ* = −1. While the basis of T and K can be constructed using the same procedure for all finite connected graphs *G*0, for the last subspace, we provided a construction based on case separation, depending on if the graph is bipartite or not and if it involves self-loops.

The use of fundamental cycles allowed us to considerably expand the results previously found in the literature, which are often limited to planar graphs. The derived construction of the attractor space enables better understanding of the quantum transport models on graphs. In addition, our results reveal that the attractor space can contain subspaces of eigenvalues different from *λ* = ±1. In such a case, the evolution of the Grover walk with sink will have more complex asymptotic cycle. In fact, the example presented in Section 5 exhibits an infinite asymptotic cycle, since the phase *θ* of the eigenvalues *λ*± = ±1 is not a rational multiple of *π*. This feature is missing, e.g., in the Grover walk on dynamically percolated graphs with sinks, where the evolution converges to a steady state.

**Author Contributions:** Conceptualization, E.S.; formal analysis, E.S., M.Š., N.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** E.S. acknowledges financial supports from the Grant-in-Aid of Scientific Research (C) No. JP19K03616, Japan Society for the Promotion of Science and Research Origin for Dressed Photon. M.Š. is grateful for the financial support from MŠMT RVO 14000. This publication was funded by the project "Centre for Advanced Applied Sciences", Registry No. CZ.02.1.01/0.0/0.0/16\_019/0000778, supported by the Operational Programme Research, Development and Education, co-financed by the European Structural and Investment Funds and the state budget of the Czech Republic.

**Conflicts of Interest:** The authors declare no conflict of interest.
