The Hermitian adjacency matrix of a mixed graph with no loops and at most one edge or arc between each pair
of vertices has entries 
for an undirected edge,
for an arc 
,
for an arc 
,
and 
otherwise. Here 
is the imaginary unit. This
convention makes 
a Hermitian matrix, so
all its eigenvalues are real (Song and Lin 2026).
For an undirected graph, 
is the ordinary adjacency
matrix. A single arc 
instead gives
 |
Its eigenvalues are 
and 
. The matrix exponential 
is a unitary matrix used to define perfect
state transfer on mixed graphs.
