Basic graph definitions

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Directed and undirected graphs

  1. A directed graph [math]\displaystyle{ G=(V,A) }[/math] consists of a finite set [math]\displaystyle{ V }[/math] of nodes (a.k.a. vertices) and a multiset [math]\displaystyle{ A }[/math] of ordered pairs of nodes. The elements of [math]\displaystyle{ A }[/math] are the arcs of [math]\displaystyle{ G }[/math].
  2. An undirected graph [math]\displaystyle{ G=(V,E) }[/math] consists of a finite set [math]\displaystyle{ V }[/math] of nodes (a.k.a. vertices) and a multiset [math]\displaystyle{ E }[/math] of unordered pairs of nodes, the edges of [math]\displaystyle{ G }[/math].
  3. A directed or undirected graph is simple, if:
    1. No node is paired with itself in [math]\displaystyle{ A }[/math] and [math]\displaystyle{ E }[/math], respectively.
    2. The multisets [math]\displaystyle{ A }[/math] and [math]\displaystyle{ E }[/math], respectively, are sets.
  4. A directed graph [math]\displaystyle{ G=(V,A) }[/math] is symmetric if [math]\displaystyle{ (v,w)\in A }[/math] implies [math]\displaystyle{ (w,v)\in A }[/math], and anti-symmetric if [math]\displaystyle{ (v,w)\in A }[/math] implies [math]\displaystyle{ (w,v)\not\in A }[/math].

Adjacency, incidence, and degree

  1. Two nodes of a directed or undirected graph are called adjacent if they share at least one arc/edge.
  2. A node and an arc/edge are called incident if the node belongs to the arc/edge.
  3. For a node, the number of incident arcs/edges is the degree of this node.
  4. Consider a directed graph [math]\displaystyle{ G=(V,A) }[/math] and a node [math]\displaystyle{ v\in V }[/math]:
    1. An arc [math]\displaystyle{ (v,w)\in A }[/math] is an outgoing arc of [math]\displaystyle{ v }[/math], and an arc [math]\displaystyle{ (w,v)\in A }[/math] is an incoming arc of [math]\displaystyle{ v }[/math].
    2. The outdegree of [math]\displaystyle{ v }[/math] is the number of its outgoing arcs, the indegree of [math]\displaystyle{ v }[/math] is the number of its incoming arcs.

Representations of graphs

We focus on directed graphs because an undirected graph is usually represented as a symmetric directed graph as introduced above. Let and . We denote and .

  1. Adjacency matrix: an -matrix such that, for ,
if , and
if .
  1. Incidence matrix: an -matrix such that, for and , it is
if  for some ,
if  for some ,

, otherwise.

  1. Incidence list: a sequence of length . For , is a sequence of arcs and contains all outgoing arcs of .

Connectedness

An undirected graph is said to be connected if, for each pair of nodes, there is a path connecting this pair. A directed graph is said to be weakly connected if, for each pair of nodes, there is a generalized path connecting this pair. A directed graph is said to be strongly connected if, for each pair of nodes, there is a path connecting this pair.