Show that for a family of ε-vertex expanders the expansion parameter $h(G_j )$ stays bounded away from 0. Conversely, let $G_1$, $G_2$, . . . be a sequence of k-regular graphs whose number of vertices goes to $\infty$. Show that if the expansion parameter satisfies $h(G_j )$ ≥ $γ$ for some $γ$ > 0, then $G_1$, $G_2$, . . . is a family of ε-vertex expanders for some ε > 0.
The definition of ε-vertex expanders I am using is as following: a sequence $G_1$, $G_2$, . . . of k-regular graphs on vertex sets $V_1$, $V_2$, . . . is called a family of ε-vertex expanders if |$V_j$ | goes to infinity and for all sets S ⊂ Vj of at most half the vertices, we have |$N(S) \cup S$| ≥ (1 + ε)|$S$|.
I know that a d regular graph that has the largest two eigenvalues, say $\lambda_1$,$\lambda_2$, with $\lambda_1$ bigger than $\lambda_2$, then expansion parameter $h(G) \ge$ $\frac 1 2 (\lambda_1$-$\lambda_2)$, and obviously $ (\lambda_1-\lambda_2) \ge 0$. So is this sufficient to prove the first direction because if it is true for every $G_i$, then it is true for the family of G? For the other direction, I am stuck on how to start the proof (i tried to use the definition of expansion parameter but do not know how to apply it here). Any help is appreciated.
Expansion is covered in detail in the book "Pseudorandomness" in the chapter "Expanders" by Vadhan: https://people.seas.harvard.edu/~salil/pseudorandomness/expanders.pdf.
You have asked two perliminary questions on expander graphs recently. I suggest reading up on some literature. This chapter is a good place to start! It includes comparisons between the three main types of expansion: vertex, edge and spectral. See in particular Theorems 4.6, 4.9 and 4.14.