Resistance distance in connected balanced digraphs

Let $D=(V,E)$ be a strongly connected and balanced digraph with vertex set $V$ and arc set $E\mathrm{.}$ 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 $D\mathrm{.}$ Let $L$ be the Laplacian matrix of $D$ and $L^{†}=(l_{ij}^{†})$ be the Moore-Penrose inverse of $L\mathrm{.}$ 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 $i\mathrm{\ne }j$ and with $r_{ij}^{D_t} \leq d_{ij}^{D_t} \ \forall
t = 1 \ $\mathrm{t}\mathrm{o}$ \ k.$Let$$\mathcal{C}$$ be a collection of connected, balanced digraphs, each member of which is a finite union of the form $D_1\cup D_2\cup \mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\cup D_k$ where each $D_i$ is a connected and balanced digraph with $D_{i} \cap ( D_1 \cup D_2 \cup …\cup D_{i-1} )$beingasinglevertex,forall$i,$ $1 < i \leq k.$Inthispaper,weshowthatforanydigraph$D$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 ).