A mixed graph is a graph in which both directed edges and undirected edges may occur.
If only directed edges exist, the graph is called a directed graph.
If only undirected edges exist, it is called an undirected graph.
For a mixed graph with no loops and at most one undirected edge or graph
arc between each pair of vertices, the Hermitian
adjacency matrix represents an undirected edge
by 1 and a graph arc by the conjugate
pair 
and 
.
This extends spectral questions such as perfect
state transfer to mixed graphs (Song and Lin 2026).
A different question concerns the smallest possible vertex count for prescribed undirected degree 
, indegree
and outdegree 
, and girth 
. A mixed cage attains
this minimum 
.
