Degree in a multigraph, indegree and outdegree in a digraph, and their underlying connectivity

Let $M$ be a finite undirected multigraph (Multigraphs, loops and directed graphs as variants distinct from the default finite simple graph). The degree ${\mathrm{deg}}_M(v)$ of a vertex $v$ is the number of incident edge ends: each nonloop edge incident with $v$ contributes one, and each loop at $v$ contributes two.

For a finite digraph $D=(V,A)$, the outdegree and indegree are

${\mathrm{deg}}_D^{+}(v)=\mathrm{\mid }\{(v,w)\in A:w\in V\}\mathrm{\mid },{\mathrm{deg}}_D^{-}(v)=\mathrm{\mid }\{(w,v)\in A:w\in V\}\mathrm{\mid }\mathrm{.}$

A directed loop $(v,v)$ contributes one to each quantity. A digraph vertex is isolated when both degrees are zero. A multigraph vertex is isolated when its degree is zero.

The underlying undirected graph of a multigraph is obtained by deleting loops and replacing each nonempty parallel class by one simple edge. The underlying undirected graph of a digraph has an edge $\{u,v\}$ for distinct $u,v$ whenever $(u,v)$ or $(v,u)$ is an arc. A multigraph or digraph is weakly connected when this underlying graph is connected in the sense of Connected graphs and connected components defined by the existence of vertex paths. Thus the null graph is not weakly connected and a one-vertex underlying graph is connected.

These finite counts use The cardinality $\mid A\mid$ of a finite set, and the underlying objects use the simple-graph convention of A finite simple graph is a finite vertex set together with a set of two-element vertex subsets.