  Given a graph $G$, the exponential distance matrix is defined entry-wise by
letting the $(u,v)$-entry be $q^{\text{dist}(u,v)}$, where $\text{dist}(u,v)$
is the distance between the vertices $u$ and $v$ with the convention that if
vertices are in different components, then $q^{\text{dist}(u,v)}=0$. In this
paper, we will establish several properties of the characteristic polynomial
(spectrum) for this matrix, give some families of graphs which are uniquely
determined by their spectrum, and produce cospectral constructions.
