TITLE: The first eigenvalue of a graph - what does it reflect?
QUESTION [17 upvotes]: A big-picture question: what "physical properties" of a graph, and in particular of a bipartite graph, are encoded by its largest eigenvalue? If $U$ and $V$ are the partite sets of the graph, with the corresponding degree sequences $d_U$ and $d_V$, then it is easy to see that the largest eigenvalue $\lambda_{\max}$ satisfies
  $$ \sqrt{\|d_U\|_2\|d_V\|_2} \le \lambda_{\max} \le \sqrt{\|d_U\|_\infty\|d_V\|_\infty}; $$
in particular, if the graph is $(r_U,r_V)$-regular, then $\lambda_{\max}=\sqrt{r_Ur_V}$. (A reference, particularly for the double inequality above, will be appreciated.) In the general case, the largest eigenvalue also reflects in some way the "average degree" of a vertex - but is anything more specific known about it? To put it simply,

What properties of a (bipartite) graph can be read from its largest eigenvalue?


A brief summary and common reply to all those who have answered so far.

Thanks for your interest and care!
To make it very clear: I am interested in the usual, not Laplacian eigenvalues.
Although the largest eigenvalue is related to the average degree, for non-regular graphs this does not tell much; hence, I believe, understanding the meaning of the largest eigenvalue in terms of the "standard" properties of the graph is of certain interest.
It is true that different bipartite graphs (as $K_{1,ab}$ and $K_{a,b}$) may have the same largest eigenvalue, but, I believe, this does not mean that the largest eigenvalue cannot be suitably interpreted.  
I still could not find a reference to the displayed inequality above. (@kimball: Lovasz does not have it.)

REPLY [3 votes]: Assume the graph is connected.
Let $\left| 1 \right>$ be the vector of all 1's (in Dirac notation), and let $A$ be the adjacency matrix.  Then $\left<1|A|1\right>$ is the number of edges of the graph (well, actually twice the number of edges).  Similarly, $\left<1|A^n|1\right>$ is the number of paths of length $n$, where a path is a sequence of vertices with consecutive vertices connected, repetitions allowed in the case of loops.  Call this $P_n$.
Since the graph is connected, its adjacency matrix is irreducible and by the Perron-Frobenius theorem the first eigenvalue is simple and the eigenvector $\left| v \right>$ has positive components.  Therefore, $\left< v | 1 \right> > 0$, allowing use of the power iteration method.  As $n \to \infty$, $A^n\left|1\right>$ approaches the first eigenvector of $A$ (aside from normalization).  So,
$$
\lim_{n \to \infty} \frac{P_{n+1}}{P_n} =
\lim_{n \to \infty} \frac{P_{2n+1}}{P_{2n}} =
\lim_{n \to \infty} \frac{\left<1|A^n A A^n|1\right>}{\left<1|A^n A^n|1\right>} = \lambda_{\textrm{max}}.
$$
The largest eigenvalue then tells how the number of paths of length $n$ grows, as $n$ grows (keeping in mind that I used a nonstandard definition for "path").  Furthermore, the limit written above approaches $\lambda_\textrm{max}$ from below.
Theorem 4 of Yu, Lu, Tian (2004) is equivalent to $\lambda_\textrm{max} \ge \sqrt{P_4 / P_2}$, but is expressed in terms of the degrees and 2-degrees of the vertices.