The exponential distance matrix of bi-block graphs

Let $G$ be a connected graph with vertex set $\{v_1, v_2, \ldots, v_\mathbf{n}\}$. As a variant of the classical distance matrix, the \emph{exponential distance matrix} was introduced independently by Yan and Yeh, and by Bapat et al. For a nonzero indeterminate $q$, the exponential distance matrix $\mathscr{F} = (\mathscr{F}_{ij})_{\mathbf{n} \times \mathbf{n}}$ of $G$ is defined by $\mathscr{F}_{ij} = q^{d_{ij}},$ where $d_{ij}$ denotes the distance between vertices $v_i$ and $v_j$ in $G$. A connected graph is said to be a \emph{bi-block graph} if each of its blocks is a complete bipartite graph, possibly of varying bipartition sizes. In this paper, we obtain explicit expressions for the determinant, inverse, and cofactor sum of the exponential distance matrix of bi-block graphs. As a consequence, some known results concerning the exponential distance matrix and the $q$-Laplacian matrix are generalized.