---- > [!definition] Definition. ([[component of a graph]] — **Undirected**) > A **component of an undirected graph** is a subset of nodes such that there exists at least one [[parameterized curve|path]] between pairwise members, and no other node from the [[graph]] can be appended to the subset while preserving this property. > \ > A [[graph]] has one **component** iff it is a [[connected graph]]. A [[graph]] with more than one **component** is a **disconnected graph**. > \ > The [[adjacency matrix]] of a **disconnected graph** is [[block matrix|block]] [[diagonal matrix|diagonal]] provided that nodes of each component are indexed sequentially relative to the other components. > [!definition] ([[component of a graph]] — **Directed**) > A **weakly connected component** of a [[network|directed graph]] is a [[component of a graph|component]] of the [[graph]] when it is viewed as undirected. > \ Two nodes $a$ and $b$ in a [[network|directed graph]] are said to be **strongly connected** if there exists a [[parameterized curve|path]] from $a \to b$ and a [[parameterized curve]] from $b \to a$. \ > A **strongly connected component** of a [[network|directed graph]] is a [[component of a graph|component]] of the [[network|undirected]] [[graph]] created by connecting two edges iff they are **strongly connected**. ![[CleanShot 2023-09-17 at [email protected]]] An **out-component** of a [[network|directed graph]] is the set of nodes that are reachable via directed paths from a specified node $a$, and including $a$ itself. Note that this is a property both of the network structure and starting node (really, the strongly connected component containing the starting node) — not the network structure alone. \ An **in-component** of a [[network|directed graph]] is the set of nodes from which there is directed path to a specified node $a$, including $a$ itself. ![[CleanShot 2023-09-17 at [email protected]]] > [!basicexample] > - For any [[primitive matrix]] $A$, the corresponding (weighted) [[network|directed graph]] is [[component of a graph|strongly connected]]. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```