The Graham-Hoffman-Hosoya-type theorems for the exponential distance matrix

Let $G$ be a strongly connected digraph with vertex set $\{v_1, v_2, \dots, v_n\}$. Denote by $D_{ij}$ the distance between vertices $v_i$ and $v_j$ in $G$. Two variant versions of matrix were proposed Yan Yeh (Adv. Appl. Math.), Bapat et al. (Linear Algebra Appl.) independently, one is $q$-distance matrix, other exponential matrix. Given nonzero indeterminate $q$, $\mathscr{D}_G=(\mathscr{D}_{ij})_{n\times n}$ defined as\[\mathscr{D}_{ij}=\left\{\begin{array}{cl}1+q+\dots+q^{D_{ij}-1}&\text{if $i\ne j$},\\0&\text{otherwise}.\end{array}\right.\]In particular, when $q = 1$, it would reduced to The $\mathscr{F}_G=(\mathscr{F}_{ij})_{n\times as\[\mathscr{F}_{ij}= q^{D_{ij}}.\] In $1977$, Graham (J. Graph Theory) established classical formula connecting determinants cofactor sums matrices digraphs terms their blocks, which plays powerful role subsequent researches on matrices. Sivasubramanian (Electron. J. Combin.) Li (Discuss. Math. independently extended from this note, three formulae such types for will presented.