Spectral properties of the exponential distance matrix

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