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

Abstract

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 the distance matrix were proposed by Yan and Yeh (Adv. Appl. Math.), and Bapat et al.  (Linear Algebra Appl.) independently, one is the $q$-distance matrix, and the other is the exponential distance matrix. Given a nonzero indeterminate $q$, the $q$-distance matrix $\mathscr{D}_G=(\mathscr{D}_{ij})_{n\times n}$ of $G$ is 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 be reduced to the distance matrix of $G$. The exponential distance matrix $\mathscr{F}_G=(\mathscr{F}_{ij})_{n\times n}$ of $G$ is defined as
\[
\mathscr{F}_{ij}= q^{D_{ij}}.
\] In $1977$, Graham et al.  (J. Graph Theory) established a classical formula connecting the determinants and cofactor sums of the distance matrices of strongly connected digraphs in terms of their blocks, which plays a powerful role in the subsequent researches on the determinants of distance matrices. Sivasubramanian (Electron. J. Combin.) and Li  et al. (Discuss. Math. Graph Theory) independently extended it from the distance matrix to the $q$-distance matrix. In this note, three formulae of such types for the exponential distance matrices of strongly connected digraphs will be presented.