respectively. $\Re(\varphi(C)) $ is the real part of the gain of a directed cycle $ \overrightarrow{C}$.

Next result gives a bound for the spectral radius of the adjacency matrix $A(G)$ of a simple graph $G$ in terms of the largest vertex degree of $G$.

\[Theo.11\] Let $ G $ be a connected graph with $ n $ vertices and $ m $ edges. Then $ \rho(G) \leq \Delta$. Furthermore, equality holds if and only if $ G $ is regular.

A digraph is said to be *weakly connected* if $ \Gamma(X) $ is connected. The following extension of the above result for the Hermitian adjacency matrices is known.

\[Theo.2.6\] Let $ X $ be a digraph and $ \Delta $ be the largest vertex degree of $ \Gamma(X) $. Then $ \rho(H(X))\leq \Delta $. If $ X $ is weakly connected, then equality holds if and only if $ \Gamma(X) $ is $ \Delta $-regular and there is a partition of vertex set of $ X $ into four parts( possibly empty) $ V_{1}, V_{-1}, V_{i} $, and $ V_{-i} $ such that one of the following holds:

1.  For each $ j \in \{ \pm 1, \pm i\} $, 