Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Line graphs of Multi-Graphs and the forbidden graph $E_6$   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 $E_6$.

25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
29
The categories of this paper are: funct-an
14
The categories of this paper are: math-ph
5
The categories of this paper are: gr-qc
12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas
0
The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
7
The categories of this paper are: hep-th
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
33
The categories of this paper are: plasm-ph