  Let X be a symmetric graph of type k and order r, where k,r $\ge$ 2 are
integers. In this paper we give explicite expressions of the horocyclic Abel
transform and its dual, as well as their inverses X. We then derive the
Plancherel measure for the Helgason-Fourier transform on G and give a version
of the Kunze-Stein phenomenon thereon. Finally, we compute the solution to the
shifted wave equation on X, using {\`A}sgeirsson's mean value theorem and the
inverse dual Abel transform.
