  We establish a new formula for the heat kernel on regular trees in terms of
classical I-Bessel functions. Although the formula is explicit, and a proof is
given through direct computation, we also provide a conceptual viewpoint using
the horocyclic transform on regular trees. From periodization, we then obtain a
heat kernel expression on any regular graph. From spectral theory, one has
another expression for the heat kernel as an integral transform of the spectral
measure. By equating these two formulas and taking a certain integral
transform, we obtain several generalized versions of the determinant formula
for the Ihara zeta function associated to finite or infinite regular graphs.
Our approach to the Ihara zeta function and determinant formula through heat
kernel analysis follows a similar methodology which exists for quotients of
rank one symmetric spaces.
