Line graphs of Multi-Graphs and the forbidden graph $E_6$

TL;DR

This paper characterizes line graphs of multi-graphs by identifying forbidden subgraphs related to the $E_6$ Weyl group, linking graph theory with algebraic structures in geometry.

Contribution

It provides new characterizations of line graphs of multi-graphs using forbidden subgraphs connected to $E_6$ geometry.

Findings

A graph is a line graph iff it avoids 33 specific forbidden graphs.

These forbidden graphs correspond to bases of anisotropic vectors in $E_6$ geometry.

The results connect graph theory with algebraic and geometric structures.

Abstract

The line graph $\Gamma$ of a multi-graph $\Delta$ is the graph whose vertices are the edges of $\Delta$, where two such edges are adjacent if and only if they meet in a single vertex of $\Delta$. We provide several characterizations of such line graphs and in particular show that a graph is a line graph if and only if it does not contain one of $33$ graphs, all of which correspond to bases of anisotropic vectors of a $6$-dimensional orthogonal geometry of $-$-type over a field with two elements, or, equivalently, to sets of $6$ generating reflections in the Weyl group of type $E_6$.