I'm currently trying to give a simple argument why the Petersen graph is not a generalized line graph (it also is an exercise in Godsil/Royle) with following definition:

Let $\Gamma$ be a graph and $A$ its adjacency matrix. $\Gamma$ is called a generalized line graph if there exist vectors $v_1,\dots,v_s \in D_n = \{\pm e_i \pm e_j \mid 1 \leq i \not= j \leq n\} \subseteq \mathbb{R}^n$ such that $A + 2I = G$ with $G_{ij} = \left<v_i,v_j\right>$.

I know that the eigenvalues of the Petersen graph are $-2, 1$ and $3$ with multiplicities $4,5,1$ (which leads to the conlusion that the rank of $A+2I$ is $6$) and that it is the only strongly regular graph with parameters $(10,3,0,1)$.

I already checked this question: Representing Petersen graph in root system $E_6$ but it doesn't give an answer to this question.


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