Discrete Math Notes:
Chapter 5: Functions and Relations
5.8 Directed graphs, paths, and cycles
directed graph consists of a pair (V, E)
vertices, V is considered to be a set of vertices
directed edges, E is considered to be a set directed edges
vertex, An individual element of V
tail, vertex u
head, vertex v
self-loop, If the head and the tail of an edge are the same vertex it is considered to be a self-loop
in-degree, is the number of edges pointing into it
out-degree, a vertex is the number of edges pointing out of it.
Walk, from vo to v in a directed graph G is a sequence of alternating vertices and edges that starts and ends with a vertex.
length of a walk, I, the number of edges in the walk
open walk, a walk in which the first and last vertices are not the same
closed walk, walk in which the first and last vertices are the same.
trail, an open walk in which no edge occurs more than once.
Circuit, a closed walk in which no edge occurs more than once.
Path, a trail in which no vertex occurs more than once
cycle, a circuit of length at least 1 in which no vertex occurs more than once, except the first and last vertices which are the same.
Discrete Math Notes:
Chapter 5: Functions and Relations
5.9 Composition of relations
Composition, is another relation on A, denoted S o R. The pair (a, c) E S o R if and only if there is a b E A such that (a, b) E R and (b, c) E s.
Overview:
Relation R and S on set A = { a, b, c, d }
SOR
Captions ^
1. R and S are relations on set A. (a, c) E S o R because there is a vertex d such that (a,d) E R and(d,c)Es 2. Since (c,d) E R and (d,c) E S, (c,c) E S o R 3. Since (b,b) E R and (b,d) E S, (b,d) E S o R 4. Since (b,b) E R and (b,a) E S, (b,a) E S o R.
Another relation on A, labeled S o R, is the composition of relations R and S on set A. If and only if there is a b A such that (a, b) R and (b, c) E S, the pair (a, c) E S o R. Because it is natural to read an expression from left to right, the ordering of the relations R and S in the statement "S o R" may appear strange. The ordering established here for the composition of relations, on the other hand, is consistent with the ordering defined for the composition of functions, in which R is applied first, followed by S.
Discrete Math Notes:
Chapter 5: Functions and Relations
5.10 Graph powers and the transitive closure
power of G, the graph Gk is defined to be the directed graph whose edge s