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 $7 - 2\sqrt 6 \approx 2.10$.