General definitions for mixed graphs? (degrees, connectivity, etc)
The definition I found in Zhang, On the completeness of orientation rules for causal discovery in the presence of latent confounders and selection bias (Section 2.1) is:


  A mixed graph is a vertex-edge
  graph that can contain three kinds of
  edges: directed ($\rightarrow$), bi-directed
  ($\leftrightarrow$) and undirected ($-$), and at most
  one edge between any two vertices.
  
  ...
  
  Two vertices are said to be
  adjacent in a mixed graph if there is an edge (of any kind) between them.
  Given a mixed graph $G$ and two
  adjacent vertices A,B therein, A is a
  parent of B and B is a child of A if A $\rightarrow$ B is in $G$; A is called
  a spouse of B (and B a spouse of
  A) if A $\leftrightarrow$ B is in $G$; A is called a
  neighbor of B (and B a neighbor of A) if A $-$ B is in $G$. 
  
  ...
  
  A directed path from $V_{0}$
  to $V_{n}$ in $G$ is a sequence of distinct vertices
   such that for 0
  $\leq$ i $\leq$ n-1, $V_{i}$ is a parent of
  $V_{i+1}$ in $G$. A is called an
  ancestor of B and B a descendant of A if A = B or there is a directed path from A to B.


Richardson and Spirtes give another nice and simple definition of mixed graphs in Ancestral Graph Markov Models (Section 2.2).


  Note that the three edge types should be considered as distinct symbols, and in particular 
  
  A $-$ B $\neq$ A $\leftrightarrows$ B $\neq$ A $\leftrightarrow$ B.


However, none of them defines in and out degree.