Graph zeta functions and Wilson loops in a Kazakov–Migdal model

Abstract In this paper, we consider an extended Kazakov–Migdal model defined on arbitrary graph. The partition function of the model, which is expressed as summation all Wilson loops graph, turns out to be represented by Bartholdi zeta weighted unitary matrices edges cycle graph at finite N generating generalized Catalan numbers. can exactly evaluated large N, infinite product a kind deformed Ihara function. non-zero-area do not contribute leading part 1/N expansion free energy but next leading. semi-circle distribution eigenvalues scalar fields still exact solution regular it reflects only zero-area loops.