Relationship between the different Definitions

The expansion parameters defined above are related to each other. In particular, for any graph G, . Consequently, every vertex expander family is also an edge expander family.

Similarly, when G is d-regular, there is a relationship between h(G) and the spectral gap d − λ₁ of G. An inequality due to "Cheeger and Buser in the continuous case and Tanner, Alon, and Milman in the discrete case" [1] states that

As a result, a family of graphs is an edge expander family if and only if is a spectral expander family.