Lecture No. 28
For a digraph G = (V, E),
vv
in-degree(v) = out-degree(v) = |E|
VV∈∈
∑∑
where |E| means the cardinality of the set E, i.e., the number of edges.
For an undirected graph G = (V, E),
vdegree(v) = 2|E|
V
∈
∑
where |E| means the cardinality of the set E, i.e., the number of edges.
A path in directed graphs is a sequence of vertices 〈v0, v1. . . vk〉 such that (vi-1, vi) is an
edge for i = 1, 2, . . . , k. The length of the paths is the number of edges, k. A vertex w is
reachable from vertex u is there is a path from u to w. A path is simple if all vertices
(except possibly the fist and last) are distinct.
A cycle in a digraph is a path containing at least one edge and for which v0 = vk. A
Hamiltonian cycle is a cycle that visits every vertex in a graph exactly once. A Eulerian
cycle is a cycle that visits every edge of the graph exactly once. There are also “path”
versions in which you do not need return to the starting vertex.
Figure 8.5: Cycles in a directed graph
A graph is said to be acyclic if it contains no cycles. A graph is connected if every vertex
can reach every other vertex. A directed graph that is acyclic is called a directed acyclic
graph (DAG).
There are two ways of representing graphs: using an adjacency matrix and using an
adjacency list. Let G = (V, E) be a digraph with n = |V| and let e = |E|. We will assume
that the vertices of G are indexed {1, 2 . . . n}.
An adjacency matrix is a n × n matrix defined for 1 ≤ v, w ≤ n.
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