  Graham showed with Pollak and Hoffman-Hosoya that for any directed graph $G$
with strong blocks $G_e$, the determinant $\det(D_G)$ and cofactor-sum
$cof(D_G)$ of the distance matrix $D_G$ can be computed from the same
quantities for the blocks $G_e$. This was extended to trees - and in our recent
work to any graph - with multiplicative and $q$-distance matrices. For trees,
we went further and unified all previous variants with weights in a unital
commutative ring, into a distance matrix with additive and multiplicative
edge-data.
  In this work: (1) We introduce the additive-multiplicative distance matrix
$D_G$ of every strongly connected graph $G$, using what we term the
additive-multiplicative block-datum $\mathcal{G}$. This subsumes the previously
studied additive, multiplicative, and $q$-distances for all graphs. (2) We
introduce an invariant $\kappa(D_G)$ that seems novel to date, and use it to
show "master" Graham-Hoffman-Hosoya (GHH) identities, which express $\det(D_G),
cof(D_G)$ in terms of the blocks $G_e$. We show how these imply all previous
variants. (3) We show $\det(.), cof(.), \kappa(.)$ depend only on the
block-data for not just $D_G$, but also several minors of $D_G$. This was not
studied in any setting to date; we show it in the "most general"
additive-multiplicative setting, hence in all known settings. (4) We compute
$D_G^{-1}$ in closed-form; this specializes to all known variants. In
particular, we recover our previous formula for $D_T^{-1}$ for
additive-multiplicative trees (which itself specializes to a result of
Graham-Lovasz and answers a 2006 question of Bapat-Lal-Pati.) (5) We also show
that not the Laplacian, but a closely related matrix is the "correct" one to
use in $D_G^{-1}$ - for the most general additive-multiplicative matrix $D_G$
of each $G$. As examples, we compute in closed form $\det(D_G), cof(D_G),
\kappa(D_G), D_G^{-1}$ for hypertrees.
