The following definitions are taken from:

Gross, Jonathan L., and Jay Yellen. Handbook of Graph Theory. New York: CRC, 2004

unless otherwise noted.

**Weakly Connected**- A directed graph is said to be
if its underlying undirected graph is*weakly connected*.*connected*

- A directed graph is said to be
**Connected**- An undirected graph is said to be
"if there exists a walk between every pair of its vertices."*connected*

- An undirected graph is said to be
**Mutually Reachable**- "Let
*u*and*v*be vertices in a digraph*G*. Then*u*and*v*are said to bein*mutually reachable**G*if*G*contains both a directed*u*-*v*walk and a directed*v*-*u*walk. Every vertex is regarded as reachable from itself (by the trivial walk)."

- "Let
**Strongly Connected**- "A digraph is
if every two vertices are*strongly connected*.*mutually reachable*

- "A digraph is
**Strong Component**- "A
of a digraph*strong component**G*is a maximal strongly connected subgraph of*G*. Equivalently, ais a subdigraph induced on a maximal set of*strong component*vertices.*mutually reachable*

- "A
**Component**- "The subgraphs of
*G*which are maximal with respect to the property of beingare called the components of*connected**G*."

- "The subgraphs of
**Graph Density**- "The density of a graph is the ratio of the number of edges and the number of possible edges." (from igraph library documentation).