the digraph induced by $ V_{j} $ in $ X $ contains only digons. Every other directed edge $ \overrightarrow{e_{st}} $ of $ X $ such that $ v_{s} \in V_{j} $ and $ v_{t} \in V_{(-i)j} $ for some $ j \in \{ \pm 1, \pm i\} $. See Figure \[fig1.1\].

2.  For each $ j \in \{ \pm 1, \pm i\} $, the digraph induced by $ V_{j} $ in $ X $ is an independent set. For each $ j \in \{ \pm 1, \pm i\} $, every directed edge with one end in $ V_{j} $ and one end in $ V_{-j} $ is contained in a digon. Every other directed edge $ \overrightarrow{e_{st}} $ of $ X $ is such that $ v_{s}\in V_{j} $ and $ v_{t} \in V_{ij} $ for some $ j \in \{ \pm 1, \pm i\} $. See Figure \[fig1.1\].

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Let $ A$ be an $n \times n$ Hermitian matrix and let the eigenvalues of $ A $ be ordered as $\lambda_{min}=\lambda_n\leq\lambda_{n-1}\leq \ldots \leq\lambda_1=\lambda_{max}$. Then $ \lambda_{min} \leq x^{*}Ax \leq \lambda_{max} $ for any vector $ x \in \mathbb{C} ^{n}$ with $x^{*}x=1$, with equality in the right-hand(respectively, left-hand) inequality 