Resistance distance in connected balanced digraphs

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 ).