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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.

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.code. (Also ( $\dfrac{k}{\lambda}$, $\dfrac{\lambda}{k}$)) should have same multiplicity

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