Ramanujan Graphs on Cosets of $PGL_2(\mathbb{F}_q)$

In this paper we study Cayley graphs on $\PGL_2(\mathbb F_q)$ mod the unipotent subgroup, the split and nonsplit tori, respectively. Using the Kirillov models of the representations of $\PGL_2(\mathbb F_q)$ of degree greater than one, we obtain explicit eigenvalues of these graphs and the corresponding eigenfunctions. Character sum estimates are then used to conclude that two types of the graphs are Ramanujan, while the third is almost Ramanujan. The graphs arising from the nonsplit torus were previously studied by Terras et al. We give a different approach here.