## Expander Graphs and Gaps between Primes

Citations: | 1 - 0 self |

### BibTeX

@MISC{Cioabă_expandergraphs,

author = {Sebastian M. Cioabă and M. Ram Murty},

title = {Expander Graphs and Gaps between Primes},

year = {}

}

### OpenURL

### Abstract

The explicit construction of infinite families of d-regular graphs which are Ramanujan is known only in the case d−1 is a prime power. In this paper, we consider the case when d − 1 is not a prime power. The main result is that by perturbing known Ramanujan graphs and using results about gaps between consecutive primes, we are able to construct infinite families of “almost ” Ramanujan graphs for almost every value of d. More precisely, for any fixed ǫ> 0 and for almost every value of d (in the sense of natural density), there are infinitely many d-regular graphs such that all the non-trivial eigenvalues of the adjacency matrices of these graphs have absolute value less than (2 + ǫ) √ d − 1. 1