A graph consists of vertices (nodes) connected by edges (arcs).
The two main categories of graphs are undirected graphs, in which edges don't have any particular direction, and directed graphs, where edges have direction - for example, there may be an edge from node A to node B, but no edge from node B to node A.
Generally, we use the following notation for a graph
Within undirected graphs there are other distinctions:
For a connected graph with no parallel edges (i.e. each pair of vertices has only zero or one edge between it),
Generally, a graph is said to be sparse if
An adjacency matrix requires
Consider the accompanying example graph.
Here the path
A cycle is a directed path which ends where it starts. Here,
A loop is any path (directed or not) which ends where it starts.
A graph with no cycles is said to be acyclic.
A chord is any edge which connects two non-adjacent nodes in a loop.
Directed acyclic graphs (DAGs) are used often. In a DAG, parents are the nodes which point to a given node; the nodes that a given node points to are its children. A family of a node is the node and its parents.
The Markov blanket of a node is its parents, children, and the parents of its children (including itself).
For an undirected graph, a node's neighbors are noes directly connected to it.
In an undirected graph, a clique is a fully-connected subset of nodes. All members of a clique are neighbors.
A maximal clique is a clique which is not a subset of another clique.
In this graph:
An undirected graph is connected if there is a path between every pair of nodes. That is, there is no isolated subgraph.
For a non-connected graph, its connected components are its subgraphs which are connected.
A singly connected graph is the connected graph (directed or not) where there is only one path between each pair of nodes. This is equivalent to a tree.
A non-singly connected graph is said to be multiply connected.
A spanning tree of an undirected graph is one where the sum of all edge weights is at least as large as any other spanning trees'.
Graphs can be represented as an adjacency matrix:
Where a non-zero value at
A clique matrix represents the maximal cliques in a graph.
For example, this clique matrix describes the following maximal clique:
A clique matrix containing only 2-node cliques is an incidence matrix: