Ramanujan graphs on cosets of \(\operatorname{PGL}_2(\mathbb F_q)\) (Q2566177)

A Ramanujan graph is a finite \(k\)-regular graph whose eigenvalues other than \(\pm k\) have absolute values at most \(2\sqrt{k-1}\). The authors study Cayley graphs on \(\text{PGL}_2(\mathbb{F}_q)\) mod the unipotent subgroup, the split torus, and the nonsplit torus. Using the Kirillov models of the representations of \(\text{PGL}_2(\mathbb{F}_q)\) of degree greater than one, they obtain explicit eigenvalues of these graphs and the corresponding eigenfunctions. They then use character sum estimates to conclude that two types of the graphs are Ramanujan, and that the third is almost Ramanujan.