1. Chapter 5 Fundamental Concept
Graph Theory
Chapter 5 Fundamental Concept
1.1 What Is a Graph?
1.2 Paths, Cycles, and Trails
1.3 Vertex Degree and Counting
1.4 Directed Graphs

2. The KÖnigsberg Bridge Problem
Graph Theory
The KÖnigsberg Bridge Problem
Königsber is a city on the Pregel river in Prussia
The city occupied two islands plus areas on both banks
Problem:
Whether they could leave home, cross every bridge exactly once, and return home. X Y Z W

3. A Model e1 e6 e2 e5 e4 e3 e7 A vertex : a region
Graph Theory
A Model
A vertex : a region
An edge : a path(bridge) between two regions e1 e2 e3 e4 e6 e5 e7 Z Y X W X Y Z W

4. General Model A vertex : an object
Graph Theory
General Model
A vertex : an object
An edge : a relation between two objects
common member
Committee 1
Committee 2

5. Graph Theory
What Is a Graph?
A graph G is a triple consisting of:
A vertex set V(G )
An edge set E(G )
A relation between an edge and a pair of vertices e1 e2 e3 e4 e6 e5 e7 Z Y X W

6. Loop, Multiple edges Loop : An edge whose endpoints are equal
Graph Theory
Loop, Multiple edges
Loop : An edge whose endpoints are equal
Multiple edges : Edges have the same pair of endpoints
Multiple
edges
loop

7. Simple Graph Simple graph : A graph has no loops or multiple edges
Graph Theory
Simple Graph
Simple graph : A graph has no loops or multiple edges
Multiple
edges
loop
It is not simple.
It is a simple graph.

8. Graph Theory
Adjacent, neighbors
Two vertices are adjacent and are neighbors if they are the endpoints of an edge
Example:
A and B are adjacent
A and D are not adjacent A B C D

9. Subgraphs A subgraph of a graph G is a graph H such that:
Graph Theory
Subgraphs
A subgraph of a graph G is a graph H such that:
V(H)  V(G) and E(H)  E(G) and
The assignment of endpoints to edges in H is the same as in G.

10. Subgraphs Example: H1, H2, and H3 are subgraphs of G G H3 H1 H2 a b c
Graph Theory
Subgraphs
Example: H1, H2, and H3 are subgraphs of G a b G c d e b a a b H3 c d c H1 H2 e d d e

11. Finite Graph, Trivial Graph
Graph Theory
Finite Graph, Trivial Graph
Finite graph : an graph whose vertex set and edge set are finite
Trivial graph : the graph with one vertex and no edges, i.e. a single point.

12. Complement Complement of G: The complement G’ of a simple graph G :
Graph Theory
Complement
Complement of G: The complement G’ of a simple graph G :
A simple graph
V(G’) = V(G)
E(G’) = { uv | uv E(G) } G’ u v w x y u v w x y G

13. Graph Theory
Bipartite Graphs
A graph G is bipartite if V(G) is the union of two disjoint independent sets called partite sets of G
Also: The vertices can be partitioned into two sets such that each set is independent
Matching Problem
Job Assignment Problem
Workers
Boys
Girls
Jobs

14. Graph Theory
Walks, Trails1.2.2
A walk : a list of vertices and edges v0, e1, v1, …., ek, vk such that, for 1  i  k, the edge ei has endpoints vi-1 and vi .
A trail : a walk with no repeated edge.

15. Graph Theory
Path and Cycle
Path : a sequence of distinct vertices such that two consecutive vertices are adjacent
Example: (a, d, c, b, e) is a path
(a, b, e, d, c, b, e, d) is not a path; it is a walk
Cycle : a closed Path
Example: (a, d, c, b, e, a) is a cycle a b c e d

16. Connected and Disconnected
Graph Theory
Connected and Disconnected
Connected : There exists at least one path between two vertices
Disconnected : Otherwise
Example:
H1 and H2 are connected
H3 is disconnected a b d e a b c d c d H3 H1 H2 e

17. Adjacency, Incidence, and Degree
Graph Theory
Adjacency, Incidence, and Degree
Assume ei is an edge whose endpoints are (vj,vk)
The vertices vj and vk are said to be adjacent
The edge ei is said to be incident upon vj
Degree of a vertex vk is the number of edges incident upon vk . It is denoted as d(vk) ei vj vk

18. Adjacency matrix Let G = (V, E), |V| = n and |E|=m
Graph Theory
Adjacency matrix
Let G = (V, E), |V| = n and |E|=m
The adjacency matrix of G written A(G), is the n-by-n matrix in which entry ai,j is the number of edges in G with endpoints {vi, vj}.
w x y z
wxyz a b c d e w x y z

19. Incidence Matrix Let G = (V, E), |V| = n and |E|=m
Graph Theory
Incidence Matrix
Let G = (V, E), |V| = n and |E|=m
The incidence matrix M(G) is the n-by-m matrix in which entry mi,j is 1 if vi is an endpoint of ei and otherwise is 0. w
a b c d e b
wxyz y a c z e d x

20. Graph Theory
Special Graphs
Complete Graph : a simple graph whose vertices are pairwise adjacent. [ A graph is complete if each vertex is connected to every other vertex.]
The complete graph with n vertices is denoted by Kn
Complete Graph

21. Complete Bipartite Graph or Biclique
Graph Theory
Complete Bipartite Graph or Biclique
Complete Bipartite Graph

22. Graph Theory
Paths
A u,v-walk or u,v-trail has first vertex u and last vertex v; these are its endpoints.
A u,v-path: a u,v-trail with no repeated vertex.
The length of a walk, trail, path, or cycle is its number of edges.
A walk or trail is closed if its endpoints are the same.

23. Graph Theory
Even Graph, Even Vertex1.2.24
An even graph is a graph with vertex degrees all even.
A vertex is odd [even] when its degree is odd [even].

24. Regular G is regular if (G ) =  (G )
Graph Theory
Regular
G is regular if (G ) =  (G )
G is k-regular if the common degree is k.
The neighborhood of v, written Ng (v ) or N (v ) is the set of vertices adjacent to v.
3-regular