basics of graphs theory
TRANSCRIPT
GRAPHS
Mariya BohraSYMCAK.K.Wagh Institute of Engineering and Education Research
What’s there in a Graph?
Graphs consist of points called vertices lines called edges
1. Edges connect two vertices.2. Edges only intersect at vertices.3. Edges joining a vertex to itself are called
loops.
Example 1:
The following picture is a graph.List its vertices and edges.
A
E
D
C
B
Why would you use Graph?
Graphs are a structure for describing relationships between objects.
The vertices denote the objects and the edges represent the relationship.
Example 2:This is also a graph.
The vertices just
happen to have
people’s names.
Such a graph could
represent friendships
(or any kind of
relationship).
Ray Mary Suze
Jake Fanny Lulu
Luke
GRAPH TERMINOLOGIES
GRAPHS could be…• UNDIRECTED
Edges do not have a direction.
The edges indicate a two-way relationship.
Each edge can be traversed in both directions.
• DIRECTED
Edges have direction.The edges indicate
a one-way relationship.Each edge can only be
traversed in a single direction
GRAPHS could be…• UNWEIGHTED
Edges have NO weight.
• WEIGHTED
Edges have a weight.
GRAPHS could be…• CYCLIC
Graph contains cycles.
• ACYCLIC
Graph contains no cycles
A Complete Graph
• A complete graph is a graph where every vertex is adjacent to every other vertex.
• A complete graph on n vertices is denoted by Kn (or sometimes by K(n)).
Example 3 :Which ones are complete graphs?
1. 2.
3. 4.
The word ‘Adjacent’
• Adjacent Vertices are two vertices that are joined by an edge.
• Adjacent Edges are two edges that intersect at a vertex.
Example 4: 1. List out the
pairs of adjacent vertices.
2. List out the pairs of adjacent edges.
Degree of a Vertex
The degree of a vertex is the number of edges incident at that vertex, with loops counted twice.Degree of the Graph
The degree of a graph is the MAXIMUM degree its vertices.
Example 5: 1. Find the degree
of each vertex .2. What is the
degree of the graph?
Odd Degree and Even Degree
• An odd vertex is a vertex of odd degree.
• An even vertex is a vertex of even degree.
Example 6: 1. Find the
vertices with• Odd degree• Even degree
PATH
A path in a graph represents a way to get from an origin to a destination by traversing edges in the graph.
LENGTH
The length of a path is the number of edges that it uses.
EXAMPLE 7: Path from Node 1 to Node 6.What is the length of the path in both the cases?
The red path is ((6,4), (4,3), (3,2), (2,5), (5,1)); it is a path in G from node 6 to node 1
The blue path is ((6,4), (4,5), (5,1)); it is also a path in G from node 6 to node 1
EXAMPLE 8:1. Find the path
from Node 6 to Node 1.
2. Find the path from Node1 to Node 6.
CONNECTED GRAPH
A graph is connected if any two vertices can be joined by a path. If this is not possible then the graph is disconnected.
A bridge is an edge in a connected graph whose removal makes it disconnected.
EXAMPLE 9:Is the graph connected?If yes, find the bridge.
EXAMPLE 10:Is the graph connected?
EXAMPLE 11:Is the graph connected?
Representation of Graphs
1.Adjacency Matrix Representation2.Adjacency List Representation
Adjacency Matrix Representation• A graph may be represented by a two
dimensional adjacency matrix.• If G has n = |V| vertices, let M be an n by n matrix whose entries are defined by
EXAMPLE 12: Find the Adjacency Matrix Representation of the given graph
SOLUTION: Adjacency Matrix Representation of the given graph
Adjacency List Representation
The adjacency list structure is simply a linked version of the adjacency table.
EXAMPLE 13: Find the Adjacency List Representation of the given graph
SOLUTION: Adjacency List Representation of the given graph
Questions?
THANK YOU