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).

<cite authors="Landau, Zeph; Russell, Alexander">_Landau, Zeph; Russell, Alexander_, [**Random Cayley graphs are expanders: a simple proof of the Alon-Roichman theorem**](http://www.emis.de/journals/EJC/Volume_11/Abstracts/v11i1r62.html), Electron. J. Comb. 11, No. 1, Research paper R62, 6 p. (2004). [ZBL1053.05060](https://zbmath.org/?q=an:1053.05060).</cite>

There is also A. Nilli's simple proof of the Alon-Boppana theorem (4 pages) 

<cite authors="Nilli, A.">_Nilli, A._, [**On the second eigenvalue of a graph**](http://dx.doi.org/10.1016/0012-365X(91)90112-F), Discrete Math. 91, No.2, 207-210 (1991). [ZBL0771.05064](https://zbmath.org/?q=an:0771.05064).</cite>