An overview of harmonic analysis and the shifted wave equation on symmetric graphs

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.