# Representing Petersen graph in root system $E_6$

It is well-known that Petersen graph is an strongly regular graph with parameters (10,3,0,1) and can be considered as complement graph of $L(K_5)$ and its spectrum is $\{3,1^5,(-2)^4\}$.

Also, It is proved that any graph with $\lambda(G)\ge -2$ can be represented in one of the root systems $A_n$, $D_n$, $E_8$, $E_7$ or $E_6$. Also it is proved that all exceptional graphs are representable in the root system $E_8$, which is contained of $E_7$ and $E_6$.

It can be easily proved that Petersen graph is not a generalized line graph and so it is an exceptional graph. So it have a representation in the root systems $E_8$, $E_7$ or $E_6$.

I want to find a representation for this graph in $E_6$ but because I don't have enough vision in the elements of the root system $E_6$, I could not find this representation.

I will be so thankful for any help and answer about finding this representation.

Consider sets of lines in $\mathbb{R}^d$, such that the angle between two distinct lines is either $60^\circ$ or $90^\circ$. Such a set is indecomposable if we cannot partition it into two non-empty subsets such that vectors in different subsets are ore orthogonal. It is star-closed if whenever if contains lines $L$ and $M$ at an angle of $60^\circ$, it also contains the unique third line in their span that is at $60^\circ$ to $L$ and $M$. Cameron, Goethals, Seidel and Seidel proved that an indecomposable star-closed set of lines (with mutual angles $60^\circ$ and $90^\circ$) is necessarily one of the root systems $A_n$, $D_n$, $E_6$, $E_7$, $E_8$. Further any set of lines at mutual angles $60^\circ$ and $90^\circ$ can be star-closed.
Now let $A$ be the adjacency matrix of the Petersen graph. Then $A+2I$ is positive semidefinite and it is therefore the Gram matrix of 10 vectors of squared-length two. Since $-2$ has multiplicity four as an eigenvalue, $A+2I$ has rank six and so our 10 vectors lie in $\mathbb{R}^6$. It follows that the star-closure of these lines is a root system in $\mathbb{R}^6$. Since the Petersen graph is not a generalized line graph, it must be $E_6$.
• Thank you very much for your exact answer. I have studied the two references that you mentioned in your answer, but I have another question in one of them. In constructing representation of Petersen graph in example 4.9 of the paper by Cameron et al; I don't understand this part well: In $L(K_{3,3})$ plus isolated vertex $\{e_4,e_8\}\cup\{e_i+e_j:i=1,2,3;j=5,6,7\}$, we switch with respect to $\{e_i+e_{4+i}:i=1,2,3\}$. Is this the representation of Petersen in $E_6$? – A-213 Jan 1 '15 at 21:14