Weighted zeta functions for quotients of regular coverings of graphs

AbstractLet G be a connected graph. We reformulate Stark and Terras' Galois Theory for a quotient H of a regular covering K of a graph G by using voltage assignments. As applications, we show that the weighted Bartholdi L-function of H associated to the representation of the covering transformation group of H is equal to that of G associated to its induced representation in the covering transformation group of K. Furthermore, we express the weighted Bartholdi zeta function of H as a product of weighted Bartholdi L-functions of G associated to irreducible representations of the covering transformation group of K. We generalize Stark and Terras' Galois Theory to digraphs, and apply to weighted Bartholdi L-functions of digraphs

A Quantum Field Theoretical Representation of Euler-Zagier Sums

We establish a novel representation of arbitrary Euler-Zagier sums in terms of weighted vacuum graphs. This representation uses a toy quantum field theory with infinitely many propagators and interaction vertices. The propagators involve Bernoulli polynomials and Clausen functions to arbitrary orders. The Feynman integrals of this model can be decomposed in terms of an algebra of elementary vertex integrals whose structure we investigate. We derive a large class of relations between multiple zeta values, of arbitrary lengths and weights, using only a certain set of graphical manipulations on Feynman diagrams. Further uses and possible generalizations of the model are pointed out.Comment: Standard latex, 31 pages, 13 figures, final published versio

Noncommutative zeta functions of graphs and their applications

研究成果の概要 (和文) : 本研究では，グラフに付随して定まる有向辺に，非可換な量，すなわち行列や四元数のように積の順序を逆にすると結果が異なるような量，で重みづけした重み付きグラフに対するゼータ関数を定め，非可換な量を成分にもつ行列に対する行列式（非可換行列式）などによる表現の導出など，その主要な性質の解明と関連領域への応用を行った。グラフの第１種重み付きゼータ関数と第２種重み付きゼータ関数の二つについて，これまでの理論を重みが四元数の場合へ一般化し，その成果をグラフ上の四元数量子ウォークのスペクトルの解析に応用した。また，第１種重み付きゼータ関数は，四元数を特殊な場合として含む，より一般の場合への拡張を行った。研究成果の概要 (英文) : We defined several classes of weighted zeta functions of noncommutative weighted graphs; they are considered to have symmetric directed edges that are weighted by noncommutative quantities such as matrices or quaternions. We obtained main properties of the zeta functions such as determinant expressions. We generalized the theories of first and second weighted zeta functions of graphs to the case of quaternion-weighted graphs and applied them to the analysis of the spectra for quaternionic quantum walks on graphs. We also generalized the theory of first weighted zeta functions to much more general situation that includes the case of quaternions