Size of MAXCUT from eigenvalues
Yes, there is a connection between the spectrum of the graph and the size of the maximum cut. It might be easiest to see this with the normalized graph Laplacian, $L = I - D^{-1/2}AD^{-1/2}$, where $I$ is the identity, $A$ is the adjacency matrix of the graph $G = ([n], E)$ and $D$ is the diagonal matrix defined by $D_{ii} = d_i$, the degree of vertex $i$. The largest eigenvalue of $L$ is 

$$\lambda_{n-1} = \max_x\frac{x^TLx}{x^Tx} = \max_{x}\frac{\sum_{(i, j) \in E}{(x_i - x_j)^2}}{\sum_{i}{d_i x_i^2}}.$$

Because $(x_i - x_j)^2 \leq 2x_i^2 + 2x_j^2$ by Cauchy-Schwarz, $\lambda_{n-1} \leq 2$. Equality is achieved if and only if for each edge $(i, j)$, $x_i = -x_j$, which means that $\lambda_{n-1}= 2$ only if the graph has a bipartite connected component, i.e. a connected component whose maximum cut is equal to the number of edges in the component. 

For a robust version of this result, see Luca Trevisan's paper. He shows that 

$$
\lambda_{n-1} \geq 2\frac{\text{MaxCut}}{m},
$$

(where $m$ is the number of edges of the graph) and if $\lambda_{n-1} \geq 2(1-\epsilon)$, then there exits a set of vertices $S$ and a partition $(S_1, S_2)$ of $S$ such that 

$$
|e(S_1, S_2)| \geq \frac{1}{2}(1-\sqrt{16\epsilon})\text{vol}(S),
$$
where $\text{vol}(S) = \sum_{i \in S}{d_i}$ and $e(S_1, S_2)$ is the set of edges with one endpoint in $S_1$ and the other in $S_2$.

Note that the second bound does not necessarily lower bound the max cut in a useful way: it only lower bounds the maximum cut of the set $S$ in terms of $\text{vol}(S)$. This is unavoidable: think of a graph that's the disjoint union of an edge and a large graph with small maximum cut, say a clique. The maximum cut of the graph is very close to $m/2$, which is the minimum possible value of the max cut of a graph with $m$ edges. However, $\lambda_{n-1} = 2$, the maximum possible value of $\lambda_{n-1}$. In other words, while $\lambda_{n-1}$ is a relaxation of the maximum cut (up to appropriate rescaling), the gap is arbitrarily close to 2, and a 2-approximation to max cut is trivial.

Nevertheless, in the paper Luca uses the spectral result to design a recursive approximation algorithm for max cut. This is one cool thing about the paper: it uses a relaxation that, unlike the SDP relaxation of Goemans and Wiliamson, has trivial gap, but despite that an optimal solution to the relaxation actually provides enough guidance to achieve a non-trivial approximation factor.