Expanding graphs, Ramanujan graphs, and 1-factor perturbations

Abstract

We construct $(k \pm 1)$-regular graphs which provide sequences of expanders by adding or substracting appropriate 1-factors from given sequences of $k$-regular graphs. We compute numerical examples in a few cases for which the given sequences are from the work of Lubotzky, Phillips, and Sarnak (with $k-1$ the order of a finite field). If $k+1 = 7$, our construction results in a sequence of $7$-regular expanders with all spectral gaps at least $6 - 2\sqrt 5 \approx 1.52$; the corresponding minoration for a sequence of Ramanujan $7$-regular graphs (which is not known to exist) would be $7 - 2\sqrt 6 \approx 2.10$.