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Register a new userZeta functions of periodic graphs derived from quantum walk
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We define a zeta function of a finite graph derived from time evolution matrix of quantum walk, and give its determinant expression. Furthermore, we generalize the above result to a periodic graph.
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We define a zeta function of a graph by using the time evolution matrix of a general coined quantum walk on it, and give a determinant expression for the zeta function of a finite graph. Furthermore, we present a determinant expression for the zeta function of an (infinite) periodic graph.
We consider the Ihara zeta function $zeta(u,X//G)$ and Artin-Ihara $L$-function of the quotient graph of groups $X//G$, where $G$ is a group acting on a finite graph $X$ with trivial edge stabilizers. We determine the relationship between the primes of $X$ and $X//G$ and show that $Xto X//G$ can be naturally viewed as an unramified Galois covering of graphs of groups. We show that the $L$-function of $X//G$ evaluated at the regular representation is equal to $zeta(u,X)$ and that $zeta(u,X//G)$ divides $zeta(u,X)$. We derive two-term and three-term determinant formulas for the zeta and $L$-functions, and compute several examples of $L$-functions of edge-free quotients of the tetrahedron graph $K_4$.
Zeta functions of periodic cubical lattices are explicitly derived by computing all the eigenvalues of the adjacency operators and their characteristic polynomials. We introduce cyclotomic-like polynomials to give factorization of the zeta function in terms of them and count the number of orbits of the Galois action associated with each cyclotomic-like polynomial to obtain its further factorization. We also give a necessary and sufficient condition for such a polynomial to be irreducible and discuss its irreducibility from this point of view.
We lift the classical Hasse--Weil zeta function of varieties over a finite field to a map of spectra with domain the Grothendieck spectrum of varieties constructed by Campbell and Zakharevich. We use this map to prove that the Grothendieck spectrum of varieties contains nontrivial geometric information in its higher homotopy groups by showing that the map $mathbb{S} to K(Var_k)$ induced by the inclusion of $0$-dimensional varieties is not surjective on $pi_1$ for a wide range of fields $k$. The methods used in this paper should generalize to lifting other motivic measures to maps of $K$-theory spectra.
We construct spectral zeta functions for the Dirac operator on metric graphs. We start with the case of a rose graph, a graph with a single vertex where every edge is a loop. The technique is then developed to cover any finite graph with general energy independent matching conditions at the vertices. The regularized spectral determinant of the Dirac operator is also obtained as the derivative of the zeta function at a special value. In each case the zeta function is formulated using a contour integral method, which extends results obtained for Laplace and Schrodinger operators on graphs.
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