TITLE: eigenvalue estimate of the adjacency matrix
QUESTION [5 upvotes]: The adjacency matrix of a nonempty (undirected) graph has a strictly positive largest eigenvalue $\lambda_\max$.  A very easy upper estimate for it can be obtained directly by Gershgorin's theorem:
$$
\lambda_{\max}\le \Delta\ ,
$$
where $\Delta$ is the maximal degree of the graph. Are any further estimates known?
And are there known lower estimates on the lowest eigenvalue $\lambda_\min$?

REPLY [5 votes]: A classic estimate is due to Constantine:
$$
\lambda_{\min} \geq -\sqrt{\lceil \frac{n}{2} \rceil \lfloor \frac{n}{2} \rfloor}.
$$
If $m$ is the number of edges, then
$$
\lambda_{\min} \geq - \sqrt{m}.
$$
A common generalization is
$$
\lambda_{\min} \geq -\sqrt{MaxCut(G)},
$$
where $MaxCut(G)$ is the size of a maximal bipartite subgraph.
You can find these results, with references, in the 2008 paper by Bell et al.. There are more complicated results as well, in particular using the eigenvector.
If you would like to discuss such topics, I am always interested.