Structural properties of the line-graphs associated to directed networks

The centrality and efficiency measures of an undirected network $G$ were shown by the authors to be strongly related to the respective measures on the associated line graph $L(G)$. In this note we extend this study to a directed network $\vec{G}$ and its associated directed network $\vec{L}(\vec{G})$. The Bonacich centralities of these two networks are shown to be related in a surprisingly simpler manner than in the non directed case. Efficiency is also considered and the corresponding relations established. In addition, an estimation of the clustering coefficient of $\vec{L}(\vec{G})$ is given in terms of the clustering coefficient of $\vec{G}$, and by means of an example we show that a reverse estimation cannot be expected.
    Given a non directed graph $G$, there is a natural way to obtain from it a directed line graph, namely $\vec{L}(D(G))$, where the directed graph $D(G)$ is obtained from $G$ in the usual way. With this approach the authors estimate some parameters of $\vec{L}(D(G))$ in terms of the corresponding ones in $L(G)$. Particularly, we give an estimation of the norm difference between the centrality vectors of $\vec{L}(D(G))$ and $L(G)$ in terms of the Collatz-Sinogowitz index (which is a measure of the irregularity of $G$). Analogous estimations are given for the efficiency measures. The results obtained strongly suggest that for a given non directed network $G$, the directed line graph $\vec{L}(D(G))$ captures more adequately the properties of $G$ than the non directed line graph $L(G)$.