Consider the class of all possible connected simple graphs on $n$ vertices

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Consider the class of all possible connected simple graphs on $n$ vertices ($n$ is any natural number, we can choose any natural number). 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 $n$ vertices such that if $\lambda$ is an eigenvalue of $A$, then $\dfrac{1}{\lambda}$ is also an eigenvalue and if $\alpha$ is another eigenvalue (distinct from $\lambda$), then $-\dfrac{1}{\alpha}$ is an eigenvalue. Basically eigenvalues of $A$ are of the form $(\lambda,\dfrac{1}{\lambda})$ and $(\alpha,-\dfrac{1}{\alpha})$.

Basically I am trying to find the graphs for which some roots of the form $(\lambda,\dfrac{1}{\lambda})$, and some roots of the form $(\alpha,-\dfrac{1}{\alpha})$.

please help regarding this problem