TITLE: What is a "Ramanujan Graph"?
QUESTION [13 upvotes]: I noticed an apparent conflict in the definition in literature about what is a "Ramanujan graph, which I was wondering if someone could kindly clarify.
(1)
The Hoory-Linial-Wigderson review on expanders in its definition 5.11 calls a d-regular graph to be Ramanujan if the second highest (adjacency?) eigenvalue is bounded above by $2\sqrt{d-1}$.
(2)
The Batson-Spielman-Srivastava paper (arxiv:0808.0163v3) on page 2 seems to say that a d-regular graph is Ramanujan if the non-zero Laplacian eigenvalues lie between $[d-2\sqrt{d-1},d + 2\sqrt{d-1}]$.
Aren't these two things different?
Like the largest eigenvalue of a d-regular Laplacian can I guess be as large as $2d$ even for Ramanujan graphs (and will definitely be so for bipartite d-regular Ramanujan) but how come BSS is asking for a stronger upper bound than that?
And why does Hoory-Linial-Wigderson not have any lower bound in their definition? Can the smallest eigenvalue of a d-regular adjacency matrix be arbitrarily low-right?
(3)
Also the BSS paper in the footnote on page 3 says that the largest (Laplacian?) eigenvalue is at least $d+2\sqrt{d-1}$ and the second-smallest (Laplacian?) eigenvalue is at most $d-2\sqrt{d-1}$.
Are the above two bounds independent of what the Alon-Bopanna bound says as in the smallest possible value of the second-highest eigenvalue is $2\sqrt{d-1}$?

REPLY [3 votes]: Like Sebi's answer, this should really be a comment but it's too long, and easier to read with proper formatting.
The distinction between (1) and (2), once you've translated so that both talk about the same matrix, concerns whether we require one bound on the eigenvalues or two. This is a significant distinction which also caused some confusion for me (with this question), but it really just amounts to whether or not you care about "bipartiteness". For example, the rate of ordinary "forward" diffusion on a graph is governed by the smallest nontrivial eigenvalue of the Laplacian, $\lambda_2$ (the algebraic connectivity), but the rate of "reverse" diffusion is governed by the largest, $\lambda_n$ (the "bipartiteness").
Equivalently, a random walk on a bipartite graph (with $\lambda_n = 2d$) will alternate between the partite sets, and in this sense never truly converges to a uniform distribution on the vertices; what's more, for nonbipartite graphs, the closer $\lambda_n$ gets to $2d$, the more the random walk tends to alternate between two sets, corresponding to an eigenvector of $\lambda_n$. This is related to the expander mixing lemma that Sebi mentioned.
The bound you mentioned in (3) is the Alon-Boppana bound, but it is only an asymptotic bound: if you prefer, it should really be $\lambda_2 \le d - 2\sqrt{d-1} + o(1)$. So a Ramanujan graph is one which is "better than any infinite family of graphs could be".