I am wondering if there are special classes of graphs that have eigenvalue of 1 for the adjacency matrix. I know that the complete graphs, Kn, have this property, but am wondering if other graphs do as well.

Despite a lot of effort, there's no interesting characterization of graphs with 0 as an eigenvalue. I do not think as much attention has been paid to $1$, but I'd be surprised if anything useful could be said. The two problems are not unrelated: for example if $G$ is regular then it has $1$ as an eigenvalue if and only if its complement has zero as an eigenvalue. (If $G$ has $1$ as an eigenvalue with multiplicity at least two, then its complement has 0 as an eigenvalue by interlacing. 


Maybe you should ask which of the eigenvalues should have value 1. For instance when Patrick Fowler and I explored the middle eigenvalue $\lambda_n$ in the decreasing sequence of eigenvalues of a graph on $2n$ vertices, we observed that the value $1/\phi$ occurs quite frequently. We called such graphs golden graphs, since $\phi$ is the golden ratio. 


Another class of graphs with $1$ as an eigenvalue are the bipartite generalized Petersen Graphs. This includes the Desargues graph and the dodecahedron. All generalized Petersen Graphs have $1$ as an eigenvalue, so their bipartite doubles have $1$ as an eigenvalue. The definition can be found in Bollobas's Extremal Graph Theory. You can see the generalized Petersen graphs have 1 as an eigenvalue by assigning 1 to the "outer" cycle and $1$ to the other cycle to obtain an eigenvector with eigenvalue 1. 


There are a lot of graphs with this property. I just introduce two classes that are very famous: The Friendship graph $F_n$ that is $K_1\nabla nK_2$, where $\nabla$ means the join of two graphs. These graphs has $n1$ eigenvalue $1$. The second class is graph $K_n$ that is removed $1,2,3$ or $4$ edges from it. So, this class contain also five different classes!!! 


There is another interesting class (although it's probably a bit esoteric): graphs with a perfect 1code. This is shown in Lemma 9.3.4 of Algebraic Graph Theory by Godsil & Royle. 

