Note : All eigenvalues that I would referring to below would of the adjacency matrix of the graph
My question arises from having read about Expander Graphs from a few different sources. The most common definition of Expander Graphs that have seen is in terms of the spectral gap which is $d - \lambda _2$ where $\lambda _2$ is the second highest eigenvalue. However certain sources also like to define expansion in terms of $\lambda$ which is the second highest eigenvalue in absolute value. Now I am slightly confused about what is a better definition and why these two values are so mixed in literature ?
For instance Cheeger's inequalities are defined in terms of $\lambda _2$ and the Expander Mixing Lemma is defined is terms of $\lambda$ (I am not sure if there exist versions of each in terms of the other quantity). Also I have seen versions of the Alon-Bopanna bound in terms of both $\lambda$ and $\lambda _2$.
I guess my question is that is there a fundamental connection between these two quantities and is one more natural than the other to be used to define expansion ?
Also is there something fundamental that $\lambda _n$ (by $\lambda _n$ I mean the smallest eigenvalue) represent ?
I am not sure if I have been able to express my confusion clearly but I am sure I am missing something fundamental in my understanding.