  We introduce a generalized Bartholdi zeta function for simple graphs with
bounded degree. This zeta function is a generalization of both the Bartholdi
zeta function which was introduced by L.~Bartholdi and the Ihara zeta function
which was introduced by G.~Chinta, J.~Jorgenson and A.~Karlsson. Furthermore,
we establish a Bartholdi type formula of this Bartholdi zeta function for
simple graphs with bounded degree. Moreover, for regular graphs, we give a new
expression of the heat kernel which is regarded as a one-parameter deformation
of the expression obtained by G.~Chinta, J.~Jorgenson and A.~Karlsson. By
applying this formula, we give an alternative proof of the Bartholdi zeta
function formula for regular graphs.
