Maximal Line Digraphs

TL;DR

This paper investigates the maximum number of arcs in line digraphs based on the number of nodes, providing exact formulas and uniqueness conditions for maximum configurations.

Contribution

It derives explicit formulas for the maximum arcs in line digraphs and characterizes the uniqueness of such extremal structures.

Findings

Maximum arcs formula for even m: (m/2)^2 + (m/2)

Maximum arcs formula for odd m: ((m - 1)/2)^2 + m - 1

Uniqueness of maximum line digraphs for large m

Abstract

A line digraph $L(G) = (A, E)$ is the digraph constructed from the digraph $G = (V, A)$ such that there is an arc $(a,b)$ in $L(G)$ if the terminal node of $a$ in $G$ is the initial node of $b$. The maximum number of arcs in a line digraph with $m$ nodes is $(m/2)^2 + (m/2)$ if $m$ is even, and $((m - 1)/2)^2 + m - 1$ otherwise. For $m \geq 7$, there is only one line digraph with as many arcs if $m$ is even, and if $m$ is odd, there are two line digraphs, each being the transpose of the other.