# Graphs on 6 vertices with eigenvalue condition

Consider the class of all possible connected simple graphs on $6$ vertices. Now from this collection, can we find those graphs (if there exists any) satisfying the following property: Suppose that $A$ denotes the usual $(0,1)$ adjacency matrix of a graph. Now can we find those graphs explicitly (if there is any) on $6$ vertices such that if $\lambda$ is an eigenvalue of $A$, then $\dfrac{1}{\lambda}$ and $-\dfrac{1}{\lambda}$ both are eigenvalues. Simply the question can be said as: Characterize all possible simple connected graphs on $6$ vertices such that if $\lambda$ is an eigenvalue of $A$, then $\dfrac{1}{\lambda}$ and $-\dfrac{1}{\lambda}$ both are eigenvalues.

please help regarding this problem.

Follow-up question at Ask Sage question 54927.