Maximal Line Digraphs

TL;DR

This work characterizes line digraphs $L(G)$ with a fixed order $m$ that maximize the number of arcs, equivalently maximizing $\Phi(G) = \sum_{v} d_v^+ d_v^-$. It establishes exact bounds: $\Phi(G) \le (\frac{m}{2})^2 + \frac{m}{2}$ for even $m$ and $\Phi(G) \le ((m-1)/2)^2 + m - 1$ for odd $m$, then proves these bounds are tight. For $m \ge 7$, the maximal cases are uniquely realized (up to transposition) by a central-structure root digraph $O_m$ (or its transpose when $m$ is odd). The results rely on a Beineke-style forbidden-subdigraph framework, degree-based inductive arguments, and a linear-programming bound for the even-order case, providing a complete classification of extremal line digraphs. Overall, the paper advances understanding of extremal properties of line digraphs in directed graph theory.

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.

Figures (11)

• Figure 1: Forbidden subdigraphs. The subgraph Z is allowed if the dotted arc is present, which forms $K_{2,2}$.
• Figure 2: Arcs between $\Gamma_{v}^-$, $\Gamma_{v}^+$ and $\Gamma_{v}^0$
• Figure 3: Example of a line digraph with maximum number of arcs. This is a complete oriented bipartite with a maximum number of return arcs, and an additional node in the complete bipartite for the odd case.
• Figure 4: Optimal digraphs for $m = 12$ and $m = 11$. The one on the left verifies $\Phi (G) = 6*6 + (1 * 1) \cdot 6 =$42. Those on the right verify $\Phi (G) = 5 * 6 + (1 * 1) \cdot 5 + 1 * 0= 35$
• Figure 5: root digraph with four arcs and their line digraph

Theorems & Definitions (28)

• Theorem 1: beineke1968derived, Theorem 7
• Lemma 1
• proof
• Theorem 2
• proof
• Corollary 1
• Lemma 2
• proof
• Corollary 2
• proof