Regular and Vertex-Transitive Kähler Graphs Having Commutative Principal and Auxiliary Adjacency Operators

A Kähler graph is a compound of two graphs having a common set of vertices and is a discrete model of a Riemannian manifold equipped with magnetic fields. In this paper we study selfadjointness of adjacency operators of Kähler graphs and express their zeta functions in terms of eigenvalues of their principal and auxiliary adjacency operators when they are commutative. Also, we construct finite vertex-transitive Kähler graphs satisfying the commutativity condition.