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Introduction to Graph Theory

Vertices and Edges: “Networks”

Important in Computer Science, Electrical Eng., Social Sciences…

part of

Data Structures�Discrete Mathematics, Graphs and Algorithms,

part of: Combinatorics

Prof. Hassane Bouzahir

(Just google my name)

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ENSA Agadir, 2014

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1. Graphs

  • Graph

A graph G = (V, E, γ) consists of a finite set V of objects called vertices, a finite set E of objects called edges, and a function γ that assigns to each edge a subset {v, w}, where v and w are vertices (and may be the same).

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  • End points

If γ(e) = {v, w}, then vertices v and w are called the end points of e.

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1. Graphs

  • Example 1 & 2

Let V = {1, 2, 3, 4} and E = {e1, e2, e3, e4, e5}. Let γ be defined by γ(e1) = γ(e5) = {1, 2}, γ(e2) = {4, 3}, γ(e3) = {1, 3}, γ(e4) = {2, 4} .

Then G = (V, E, γ) is a graph.

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Isomorphic Graphs

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1. Graphs

  • Degree. The degree of a vertex is the number of edges having that vertex as an end point.

To be precise: the degree of a vertex is the number of endpoints on that vertex, i. e., if an edge is from a vertex to itself, we count it two times.

  • Loop. An edge is referred to as a loop if the edge is from a vertex to itself.

Note: a loop contributes 2 to the degree of a vertex, since that vertex serves as both end points of the loop.

  • Isolated. A vertex with degree 0 is called an isolated vertex.

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1. Graphs

  • Example 3�

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A: 2 B: 4

C: 1 D: 3 E: 2

a: 4 b: 2

c: 3 d: 1 e: 0

All edges with

degree 2

ENSA Agadir, 2014

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1. Graphs

  • Path in a Graph

A path п in a graph G consists of a pair (Vп, Eп) of sequences: a vertex sequence Vп : v1, v2, …, vk and an edge sequence E п : e1, e2,…,ek-1 for which

  1. Each successive pair vi, vi+1 of vertices is adjacent in G, and edge ei has vi and vi+1 as end points for i=1, …, k-1;

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  • No edge occurs more than once in the edge sequence.

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1. Graphs

  • Circuit (cycle)

A circuit is a path that begins and ends at the same vertex.

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  • Simple

A path is called simple if no vertex appears more than once in the vertex sequence, except possibly if v1= vk. In this case, the path is called a simple circuit.

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1. Graphs

  • Connected, Disconnected & Components

A graph is called connected if there is a path from any vertex to any other vertex in the graph. Otherwise, the graph is disconnected. If the graph is disconnected, the various connected pieces are called the components of the graph.

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Two components

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1. Graphs

Some important special families of graphs will be useful in our discussions.

  1. Un (discrete graph) denote the graph with n vertices and no edges.

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  • Ln (linear graph) denote the graph with n vertices and with edges {vi,vi+1} for 1≤ i <n

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U5

L2

L5

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1. Graphs

  1. Kn (complete graph) denote the graph with n vertices and with an edge {vi, vj } for all i and j.

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  • Regular

If each vertex of a graph has the same degree as every other vertex, the graph is called regular, e.g., Kn (n=2, 3, 4, 5) above.

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K3

K4

K5

K2=L2

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1. Graphs

  • Subgraph

Suppose that G = (V, E, γ) is a graph. Choose a subset E1 of the edges in E and a subset V1 of the vertices in V, so that V1 contains (at least) all the end points of edges in E1. Then H = (V1, E1, γ1) is also a graph where γ1 is γ restricted to edges in E1. Such a graph H is called a subgraph of G.

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1. Graphs

  • The subgraph Ge

Deleting one edge and no vertices

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1. Graphs

  • Example 8

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1. Graphs

  • Example 9 (Quotient graph GR)

Let G be the graph (without multiple edges) of the following figure, and let R be the equivalence relation on V defined by the partition.

{{a, m, i}, {b, f, j}, {c, g, k}, {d, h, l}}

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GR

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1. Graphs

  • Example 9

If S is also an equivalence relation on V defined by the partition.

{{i, j, k, l}, {a, m}, {f, b, c}, {d}, {g}, {h}}

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1. Graphs

  • The Quotient graph Ge

If e is an edge between vertex v and w in a graph G = {V, E, γ}, then we consider the equivalence relation whose partition consists of {v, w} and {vi}, for each vi ≠ v, vi ≠ w. (merging v and w and leaving others alone)

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1. Graphs

  • Homework

Ex. 6, Ex. 16, Ex. 20, Ex. 22, Ex. 23, Ex. 29

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