  The method of Murty and Cioab\u{a} shows how one can use results about gaps
between primes to construct families of almost-Ramanujan graphs. In this paper
we give a simpler construction which avoids the search for perfect matchings
and thus eliminates the need for computation. A couple of recent explicit
bounds on the gap between consecutive primes are then used to give the
construction of $k$-regular families with explicit lower bounds on the spectral
gaps. We then show that a result of Ben-Aroya and Ta-Shma can be improved using
our simpler construction on the assumption of the Riemann Hypothesis, which
sheds some more light on a question raised by Reingold, Vadhan and Widgerson.
