Does there exists any simple connected graph $G$ of order $n$, such that whenever $\dfrac{\lambda}{k}$ is an eigenvalue of the adjacency matrix of $G$
Does there exists any simple connected graph $G$ of order $n$, such that whenever $\dfrac{\lambda}{k}$ is an eigenvalue of the adjacency matrix of $G$, $\dfrac{k}{\lambda}$ is also an eigenvalue of adjacency matrix of $G$. Here $k$ is any positive natural number ($k=1,2,3,4,5,\ldots$) and $\lambda(\neq0)$ is an eigenvalue of adjacency matrix of $G$.
Can any one help me with a SAGE code. (Also ( $\dfrac{k}{\lambda}$, $\dfrac{\lambda}{k}$)) should have same multiplicity
For $K=1$, it is easy. But for $k=2,3,4,\ldots$, how to write a SAGE code. Please help.
See code for a similar problem at https://ask.sagemath.org/question/59220/
I tried but could not ..Can you please help me out. Where to inset that $k$?