spectrum of an adjacency matrix

A lot is known, see e.g. Andries E. Brouwer, Willem H. Haemers, Spectra of Graphs, but, from some points of view, nothing interesting :) (As usual, this depends on your particular problem.) In some subjects, it is natural to put $-2$ instead of $0$ to the diagonal, shifting the spectrum by $-2$; then, negative definite are Dybkin diagrams and semi-definite are affine Dynkin diagrams. E.g., there seems to be no satisfactory characterization of hyperbolic graphs (with a single positive eigenvalue).

Since the eigenvalues are real, and since their sum is the trace of $A$, which is zero, we see that either all eigenvalues are zero, or there are both positive and negative eigenvalues. So no non-empty graph has a positive semidefinite adjacency matrix.

I do not think there is much in the way of upper bounds on the least eigenvalue. For more regular graphs there are bounds on the size of cliques and cocliques that involve the least eigenvalues, and these can be rearranged to get upper bounds on the least eigenvalue. So if $X$ is $k$-regular on $v$ vertices and the max size of a coclique is $a$, then we have an upper bound

$$-\frac{k}{\frac{v}a -1}$$

Here I am just using the Hoffman bound on the size of a coclique.