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 $\lambda\_\min$?



eigenvalue estimate of the adjacency matrix
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.*](http://www.sciencedirect.com/science/article/pii/S0024379508001067). There are more complicated results as well, in particular using the eigenvector.


If you would like to discuss such topics, I am always interested.


