  In this paper, we examine a modification of the Kazakov-Migdal (KM) model
with gauge group $U(N_c)$, where the adjoint scalar fields in the conventional
KM model are replaced by $N_f$ fundamental scalar fields (FKM model). After
tuning the coupling constants and eliminating the fundamental scalar fields,
the partition function of this model is expressed as an integral of a graph
zeta function weighted by unitary matrices. The FKM model on cycle graphs at
large $N_c$ exhibits the Gross-Witten-Wadia (GWW) phase transition only when
$N_f > N_c$. In the large $N_c$ limit, we evaluate the free energy of the model
on a general graph in two distinct parameter regimes and demonstrate that the
FKM model generally consists of multiple phases. The effective action of the
FKM model reduces to the standard Wilson action by taking an appropriate
scaling limit when the graph consists of plaquettes (fundamental cycles) of the
same size, as in the square lattice case. We show that, for the FKM model on
such a graph, the third-order GWW phase transition universally occurs in this
scaling limit.
