1
Discrete Math
1
Graph Theory
1.1
Introduction to graphs
•
Basic definitions and properties of graphs.
•
Simple mathematical models of networks.
•
Concepts like Isomorphism as well as special types, such as connected graphs and bipartite graphs are
discussed.
1.1.1
Graph Models
•
Always look out for things or objects and relations between pairs of objects
when modeling graphs:
1.
Things
or
objects
will become
vertices
in the graphs;
2.
Relations between pairs of objects
will become
edges
of the graph.
•
A
graph
G(V,E) consists of a finite set V of
vertices
and a set E of
edges
joining different pairs of distinct
vertices.
•
Vertices a and b are
adjacent
when there is an edge (a,b).
•
Directed graphs contain
ordered
pairs of vertices, called directed edges.
•
A
directed edge
is denoted by (
g1854
g4652g1318
,c) to show the
direction from b to c.
•
A
path
P is a sequence of distinct vertices, written P=x
1
-x
2
-...-x
n
-x
1
, which
each pair of consecutive vertices
in P jointed by an edge. (p.1 notes)
•
If there is an edge (x
n
,x
1
), the sequence is called a
circuit
, written as x
1
-x
2
-...- x
n
-x
1
. (p.1 notes)
•
A graph is
connected
if there is a path between every pair of vertices.
•
The removal of certain edges or vertices from a connected graph is said to
disconnect
the graph
and the resulting graph is no longer connected.
•
If at least one pair of vertices is no longer joined by a path, then the graph is disconnected.
•
A graph in which all edges go horizontally between two sets of vertices is called
bipartite
.
•
The number of edges incident to a vertex is called the
degree of the vertex
.
•
A set C of vertices in a graph with the property that every edge is incident to at least one vertex in C is
called an
edge cover
.
•
A set of vertices without an edge between any two is called an
independent set of vertices
.

2
1.1.2
Isomorphism
•
Two graphs G and G' are called
isomorphic
if there exists a one-to-one correspondence between the
vertices in G and the vertices in G' such that a pair of vertices are adjacent in G if and only if the
corresponding pair of vertices are adjacent in G'.
•
To be isomorphic, two graphs must have the
same number of vertices
and the
same number of
edges
.
•
Degrees of vertices are preserved under isomorphism, that is,
two matched vertices must have
the same degree
.
•
Two isomorphic graphs must have the
same number of vertices of a given degree
.
•
A
subgraph
G' of a graph G is a graph formed by a
subset of vertices and edges
of G.
•
If
two subgraphs
are
isomorphic
, then
subgraphs formed by corresponding vertices and edges
must be
isomorphic
.
•
A graph with n vertices in which each vertex is adjacent to all the other vertices is called a
complete
graph
on n vertices, denoted
K
n
.

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