1. Graph Theory
Ch. 1. Fundamental Concept
1
Chapter 1
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. Graph Theory
Ch. 1. Fundamental Concept
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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.
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Ch. 1. Fundamental Concept
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A Model
❑ A vertex : a region
❑ An edge : a path(bridge) between two regions
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e6
e5
e7
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Ch. 1. Fundamental Concept
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General Model
❑ A vertex : an object
❑ An edge : a relation between two objects
common
member
Committee 1 Committee 2
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Ch. 1. Fundamental Concept
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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
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Ch. 1. Fundamental Concept
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Loop, Multiple edges
❑ Loop : An edge whose endpoints are equal
❑ Multiple edges : Edges have the same pair of
endpoints
loop
Multiple
edges
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Ch. 1. Fundamental Concept
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Simple Graph
❑ Simple graph : A graph has no loops or multiple
edges
loop
Multiple
edges
It is not simple. It is a simple graph.
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Ch. 1. Fundamental Concept
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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
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Ch. 1. Fundamental Concept
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Finite Graph, Null Graph
❑ Finite graph : an graph whose vertex set and
edge set are finite
❑ Null graph : the graph whose vertex set and
edges are empty
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Ch. 1. Fundamental Concept
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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
d
e
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Ch. 1. Fundamental Concept
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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
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Ch. 1. Fundamental Concept
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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}.
a
b
c
d
e
w
x
y z
w x y z
0 1 1 0
1 0 2 0
1 2 0 1
0 0 1 0
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Ch. 1. Fundamental Concept
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Complete Graph
❑ Complete Graph : a simple graph whose
vertices are pairwise adjacent
Complete Graph
