  Let $D = (V, E)$ be a strongly connected and balanced digraph with vertex set
$V$ and arc set $E.$ The classical distance $d_{ij}^D$ from $i$ to $j$ in $D$
is the length of a shortest directed path from $i$ to $j$ in $D.$ Let $L$ be
the Laplacian matrix of $D$ and $ L^{\dagger} = ( l_{ij}^{\dagger} )$ be the
Moore-Penrose inverse of $L.$ The resistance distance from $i$ to $j$ is then
defined by $r_{ij}^D := l_{ii}^{\dagger } + l_{jj}^{\dagger } - 2
l_{ij}^{\dagger }.$ Let $\{ D_1, D_2, ...., D_k \}$ be a sequence of strongly
connected balanced digraphs with $D_i \cap D_j$ having at most one vertex in
common for all $i \neq j$ and with $r_{ij}^{D_t} \leq d_{ij}^{D_t} \ \forall \
t = 1 \ \mathrm{to} \ k.$
  Let $\mathcal{C}$ be a collection of connected, balanced digraphs, each
member of which is a finite union of the form $D_1 \cup D_2 \cup ....\cup D_k$
where each $D_i$ is a connected and balanced digraph with $D_{i} \cap ( D_1
\cup D_2 \cup ....\cup D_{i-1} )$ being a single vertex, for all $i,$ $1 < i
\leq k.$ In this paper, we show that for any digraph $D$ in $\mathcal{C}$,
$r_{ij}^D \leq d_{ij}^D \ (*)$. This is established by partitioning the
Laplacian matrix of $D$. This generalizes the main result in [3]. As a
corollary, we deduce a simpler proof of the result in [3], namely, that for any
directed cactus $D$, the inequality (*) holds. Our results provide an
affirmative answer to a well known interesting conjecture ( cf : Conjecture 1.3
).
