My intuition on matrix exponentials, hermitian matrices, unitary matrices, adjacency matrices, and Laplacian matrices is not superb now.
For example, let $A$ be a matrix with $A^2=\mathbb I$. $A$ may be the adjacency matrix for a bijective graph; in my example it's a permutation matrix. I think then that $A$ is unitary, and we can also can relate a hermitian matrix $K$ to $A$ in the equation:
$$e^K=A\tag 1$$
as $K=\mathbb I-A.$
However, if instead we have two other permutation matrices $B$ and $C$, both of order $3$ and not of order $2$, with $B^3=C^3=BC=\mathbb I$, can we say anything about the matrix $L$ in the similar equation:
$$e^L=B+C?\tag 2$$
My guess/desire is for $L=2\mathbb I-B-C,$ but I wouldn't know how to attack this. I believe the Lie product formula would help, but I don't trust my background on these topics yet.