I was talking to my brother today, and we came up with a little conjecture. Is it true that a graph $G$ of order $n$ is connected if and only if the coefficient of $x$ in the chromatic polynomial $P(G,x)$ is nonzero? This was inspired by results like the coefficient of $x^n$ is always $1$, the constant term is always $0$, the coefficient $x^{n-1}=-|E|$, etc.
We were able to prove one direction. Suppose $G$ is disconnected. Then $G$ is the union of disconnected components, say $H$ and $K$ for simplicity's sake, and so $P(G,x)=P(H,x)P(K,x)$. But since the constant term of a chromatic polynomial is always $0$, then the least term of $P(H,x)P(K,x)$ is at least $x^2$ since the least monomial term of each is $x$, and so the coefficient of $x$ is $0$.
However, we couldn't quite complete the other direction. Our idea was to take $G$ to be a connected graph. Then $G$ has a spanning tree, with chromatic polynomial $x(x-1)^{n-1}$, which has $(-1)^{n-1}$ as its coefficient for $x$. I figured you could then recover the original graph by adding edges back, and using the fact that $P(G,x)=P(G\setminus e,x)-P(G/e,x)$ to somehow induct, but we couldn't complete the argument. So is the reverse direction true? If so, how to prove it? And if not, is there a counterexample? Thank you.