Given any polynomial p(x) over Z, can one construct a graph with characteristic polynomial p(x)? Given any polynomial $p(x)$ over $\mathbb{Z}$, can one construct a graph with characteristic polynomial $p(x)$? [Edit: Title question added to post.}
Further questions include:


*

*Are there classes of graphs that correspond to different types of polynomials
(e.g., corresponding to polynomials over finite fields? Or perhaps corresponding to certain Galois extensions of $\mathbb{Q}$?)

*If we indeed can construct this, is the graph ever unique?

*If we can't make a graph for $p(x)$ exactly, can we at least make one where we know $p(x)$ divides the characteristic polynomial?


Thanks in advance for any insights.
 A: If we consider multigraphs -- i.e., graphs with multiple edges and possibly with loops -- then there are countably infinitely many graphs on $n$ vertices and countably infinitely many degree $n$ monic polynomials in $\mathbb{Z}[t]$, so there is more of a fighting chance that every such polynomial is the characteristic polynomial of some graph than in the situation described in Gerry Myerson's answer.
I can do the case of $1$ vertex: $t-n$ is the characteristic polynomial of a multigraph iff $n \geq 0$.  
Moreover the non-negativity condition here is general: the Perron-Frobenius theorem asserts, in particular, that every matrix with non-negative real entries has at least one non-negative real eigenvalue.  
There is a large literature on eigenvalues of (multi)graphs: spectral graph theory.  I am not an expert on this (I don't even remember everything I used to know...), but there are entire texts written on the subject.  Consulting one should give some idea of what is known.
A: Let $n$ be a positive integer. Let $G$ be a graph with $n$ vertices. The adjacency matrix of $G$ is then an $n\times n$ matrix. Its characteristic polynomial is then of degree $n$. There are only finitely many graphs on $n$ vertices (up to isomorphism), but infinitely many polynomials of degree $n$ with integer coefficients. It follws that for most polynomials there is no graph for which that polynomial is the characteristic polynomial. 
