A remark on zeta functions of finite graphs via quantum walks

@article{Higuchi2014ARO,
  title={A remark on zeta functions of finite graphs via quantum walks},
  author={Yusuke Higuchi and Norio Konno and Iwao Sato and Etsuo Segawa},
  journal={Pacific Journal of Mathematics for Industry},
  year={2014},
  volume={6},
  pages={1-8}
}
From the viewpoint of quantum walks, the Ihara zeta function of a finite graph can be said to be closely related to its evolution matrix. In this note we introduce another kind of zeta function of a graph, which is closely related to, as to say, the square of the evolution matrix of a quantum walk. Then we give to such a function two types of determinant expressions and derive from it some geometric properties of a finite graph. As an application, we illustrate the distribution of poles of this… 

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References

SHOWING 1-10 OF 41 REFERENCES

Quantum walks, Ihara zeta functions and cospectrality in regular graphs

An interesting relationship between discrete-time quantum walks and the Ihara zeta function of a graph is explored and a means by which to develop zeta functions that have potential in distinguishing such structures is suggested.

A note on the discrete-time evolutions of quantum walk on a graph

For a quantum walk on a graph, there exist many kinds of operators for the discrete-time evolution. We give a general relation between the characteristic polynomial of the evolution matrix of a

Zeta Functions of Finite Graphs

Poles of the Ihara zeta function associated with a fi- nite graph are described by graph-theoretic quantities. Elementary proofs based on the notions of oriented line graphs, Perron-Frobenius

Quantum Walks on Regular Graphs and Eigenvalues

The eigenvalues of S + (U) and S +(U 2 ) for regular graphs are found and it is shown that S - (U 2) 2 + I is the transition matrix of a quantum walk on strongly regular graphs.

On the relation between quantum walks and zeta functions

We present an explicit formula for the characteristic polynomial of the transition matrix of the discrete-time quantum walk on a graph via the second weighted zeta function. As applications, we

Localization of quantum walks induced by recurrence properties of random walks

We study a quantum walk (QW) whose time evolution is induced by a random walk (RW) first introduced by Szegedy (2004). We focus on a relation between recurrent properties of the RW and localization

Quantum Random Walks in One Dimension

  • N. Konno
  • Mathematics, Physics
    Quantum Inf. Process.
  • 2002
The results show that the behavior of quantum random walk is striking different from that of the classical ramdom walk.

Coined quantum walks lift the cospectrality of graphs and trees

A Note on the Zeta Function of a Graph

The number of spanning trees in a finite graph is first expressed as the derivative (at 1) of a determinant and then in terms of a zeta function. This generalizes a result of Hashimoto to non-regular