Depending on your students' backgrounds, the following might be appropriate.
There is Landau and Russell's simple proof of the Alon-Roichman theorem (6 pages) (this one requires some representation theory but could be reasonable depending on your students' background).
Landau, Zeph; Russell, Alexander, Random Cayley graphs are expanders: a simple proof of the Alon-Roichman theorem, Electron. J. Comb. 11, No. 1, Research paper R62, 6 p. (2004). ZBL1053.05060.
There is also A. Nilli's simple proof of the Alon-Boppana theorem (4 pages)
Nilli, A., On the second eigenvalue of a graph, Discrete Math. 91, No.2, 207-210 (1991). ZBL0771.05064.